Absolutely and uniformly convergent iterative approach to inverse scattering with an infinite radius of convergence

ABSTRACT

A method and system for solving the inverse acoustic scattering problem using an iterative approach with consideration of half-off-shell transition matrix elements (near-field) information, where the Volterra inverse series correctly predicts the first two moments of the interaction, while the Fredholm inverse series is correct only for the first moment and that the Volterra approach provides a method for exactly obtaining interactions which can be written as a sum of delta functions.

RELATED APPLICATIONS

This application claims provisional priority of U.S. Provisional PatentApplication Ser. No. 60/456,175 filed 20 Mar. 2003.

ACKNOWLEDGMENT OF GOVERNMENTAL SPONSORSHIP

Portions of the research that supports the subject matter of thisapplication was supported by a grant from the National ScienceFoundation grant number CHE-0074311 and U.S. Department of Energy GrantNo. 2-7405-ENG82.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to a method for analyzing spectraincluding contributions from scattering so-called inverse scatteringanalysis using a renormalized form of the Lippmann-Schwinger equations,and to a system implemented on a computer and attached to an analyticalinstrument where a spectrum of interest is received by the instrumentand analyzed in the computer using the renormalized form of theLippman-Schwinger equations of this invention.

More particularly, the present invention relates to a method foranalyzing spectra including contributions from scattering so-calledinverse scattering analysis using a renormalized form of theLippmann-Schwinger equations, where the renormalized equation permitsabsolute and uniform convergence of the equation regardless of thestrength of interaction in the system from which the spectrum wasobtained, and to a system implemented on a computer and attached to ananalytical instrument where a spectrum of interest is received by theinstrument and analyzed in the computer using the renormalized form ofthe Lippman-Schwinger equations of this invention.

2. Description of the Related Art

Many spectral characterization include inverse scattering componentsresulting from internal reflections of an incident waveform. Theseinverse scattering components can give information on both near fieldand far field properties of the object being analyzed. However,traditional application of the Lippmann-Schwinger equations to analyzedspectra including inverse scattering components are less thesatisfactory because the Lippmann-Schwinger equations often do notconverge or give oscillatory solutions that must be truncated to productapproximate and sometimes misleading analyses.

Thus, there is a need in the art for an improved mathematical theory foranalyzing inverse scattering components that always permits solutionsbecause the equations absolutely and uniformly converge.

SUMMARY OF THE INVENTION

The present invention provides a method for analyzing inverse scatteringcomponents of a spectrum of an object of interest, where the methodutilizes equations that are absolutely and uniformly convergence andamenable efficient iterative computational determination, with leadingterms allowing for fast tentative identification of the object fromwhich the spectrum is obtained.

The present invention also provides a method for analyzing inversescattering components of a spectrum of an object of interest, where themethod utilizes equations that are absolutely and uniformly convergenceand amenable efficient iterative computational determination, withleading terms allowing for fast tentative identification of the objectfrom which the spectrum is obtained, where the method involves obtaininga reflectance and/or transmission spectra of an object of interest usingan incident waveform (electromagnetic or sonic). The spectra is thenanalyzed using the inverse scattering equations of this inventionimplemented on or in a processing unit (digital or analog) to derive apotential function representing the object. Generally, an adequatepotential function can be derived from the first few leading terms ofthe iterative solution of the equations, where few means the first fourterms, preferably the first three terms and particularly the first twoterms.

The present invention also provides an analytical instrument includingan excitation source for producing an incident waveform, a detector forreceiving either a transmission spectrum or a reflectance spectrum orboth a transmission spectrum and a reflectance spectrum of an object orvolume of interest, and a processing unit for analyzing the spectra,where the processing unit includes software encoding the inversescattering method of this invention.

The present invention also provides a sonic analytical instrumentincluding a sonic excitation source for producing an incident sonicwaveform, a detector for receiving either a sonic transmission spectrumor a sonic reflectance spectrum or both a sonic transmission spectrumand a sonic reflectance spectrum of an object or volume of interest, anda processing unit for analyzing the sonic spectra, where the processingunit includes software encoding the inverse scattering method of thisinvention.

The present invention also provides an electromagnetic analyticalinstrument including an electromagnetic excitation source for producingan incident electromagnetic waveform, a detector for receiving either anelectromagnetic transmission spectrum or an electromagnetic reflectancespectrum or both an electromagnetic transmission spectrum and anelectromagnetic reflectance spectrum of an object or volume of interest,and a processing unit for analyzing the electromagnetic spectra, wherethe processing unit includes software encoding the inverse scatteringmethod of this invention. Of course, the analytical instrument caninclude both sonic and electromagnetic excitation sources and detectors.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention can be better understood with reference to the followinadetailed description together with the appended illustrative drawings inwhich like elements are numbered the same:

FIG. 1A depicts an embodiment of an analytical instrument of thisinvention; and

FIG. 1B depicts another embodiment of an analytical instrument of thisinvention.

Referring now of FIG. 1A, an embodiment of an analytical instrument ofthis invention, generally 100, is shown to include an excitation source102 adapted to produce an incident waveform such as a sonic waveform oran electromagnetic waveform which is directed into an object or volume104 to be analyzed. The instrument 100 also includes a detector 106having a reflectance detection component 108 and a transmissiondetection component 110, where the reflectance detection component 108is adanted to detect a reflectance spectrum and the transmissiondetection component 110 is adapted to detect a transmission spectrum.The detector components 108 and 110 are connected to a processing unit112 via wires 114 and 116, where the processing unit 112 includessoftware encoding the methods of this invention.

Referring now of FIG. 1B, another embodiment of an analytical instrumentof this invention, generally 150, is shown to include an sonicexcitation source 152 adapted to produce an incident sonic waveform andan electromaanetic excitation source 154 adapted to produce an incidentelectromagnetic waveform, which are directed into an object or volume156 to be analyzed. The instrument 150 also includes a detector 158having a reflectance detection component 160 and a transmissiondetection component 162, where the reflectance detection component 160is adapted to detect a reflectance spectrum and the transmissiondetection component 162 is adapted to detect a transmission spectrum.The detector components 160 and 162 are connected to a processing unit164 via wires 166 and 168, where the processing unit 164 includessoftware encoding the methods of this invention.

We present a new inverse scattering series for quantum elasticscattering in three spherical dimensions. The new series, whichconverges absolutely, independent of the strength of the scatteringinteraction, results from a renormalization transformation of theLippmann-Schwinger Fredholm integral equation to a Volterra form. A newfeature of the formulation is that it does not require determination ofphase shifts, and it can be applied even to integral cross sectionmeasurements. The approach is illustrated by application to a simpleexample problem.

DETAILED DESCRIPTION OF THE INVENTION

The inventor has found that the inverse scattering problem encounteredin many field of spectroscopy can be solved in a more rigorous manner byapplying a simple but effective renormalization condition on thetraditional Lippmann-Schwinger equation. The simple and effectiveremomalization of the Lippmann-Schwinger equation results in a set ofequations that are absolutely and uniformally convergent for all systemsregardless of the strength of the interactions between the probingwaveform and the object being probed obviating the convergence problemsthat plague traditional Lippmann-Schwinger analyses. In fact, theinventors have found that calculation of only a limited number of termsprovides an accurate enough representation of the object being probed,either near or far field, for rapid identification. The renormalizedequations find application in all electromagnetic and sonic spectrometryapplication and is especially well-suited for analysis of data from verylow frequency electromagnetic imaging used to tract undersea objectssuch as submarines.

In this application, we introduce an approach to the inverse scatteringseries that completely solves the problem of convergence[7A]. This isachieved by renormalizing the Lippmann-Schwinger equation from aFredhohm to a Volterra structure. It was proved that the resultinginverse Born series converges absolutely and uniformly independent ofthe strength of the interaction. However, the issue of how best to dealwith the need for half-o-shell scattering information remainsoutstanding. We base our strategy for solving the inverse problem onexploiting the superior convergence properties of the Volterra-basedinverse series in combination with the introduction of aparameterization of the interaction that allows for a greatly simplifieddetermination of the interaction parameters. This enables us to takeaccount of half-o-shell (near field) effects.

Inverse Scattering Theory: Strategies Based on the Volterra InverseSeries for Acoustic Scattering

I. Introduction

It is also important to note, and we emphasize the fact, that use of aVolterra-based inversion requires either a) measurement of both thereflection and transmission amplitudes in order to achieve the simplestform of inversion b) the solution of more complicated, nonlinearalgebraic equations if only the reflection amplitude is measured. InSection II of this application, we present a simple analysis that showsclearly the need to deal with half-o-shell or near-field effects inorder to treat the problem. In Section III, we present a generalanalysis of the moments of the Fredholm and Volterra Born approximationscompared to the moments of the true interaction. This shows that theVolterra-based Born expansion yields one higher moment before thehalf-off-shell effects come into play. Then in Section IV, we considerinversion of the scattering produced by any interaction that can beexpressed as a sum of Dirac delta functions (a model interaction that isshown in more detail in Appendix B to be of practical utility). Thisinteraction also nicely illustrates the convergence properties of theVolterra series. For simplicity, we restrict ourselves to a scalarscattering wave (i.e., acoustic scattering), but the analysis is validalso for more complicated electromagnetic scattering. In Section V, wediscuss the data required for implementation of the Volterra-basedinversion. Finally, in Section VI, we give our conclusions.

II. Analysis of the Role of Half-Off-Shell Transition Amplitudes

The Born-type inverse scattering series is most simply obtained from theabstract Lippmann-Schwinger equation

$\begin{matrix}{T_{k} = {V + {{VG}_{0k}^{+}T_{k}}}} & (1)\end{matrix}$where V is the (local) interaction, T_(k) is the transition operator,and

G₀⁺is the free (unperturbed) causal Green's operator,

$\begin{matrix}{G_{0k}^{+} = \frac{k^{2}}{k^{2} - H_{0} + {i\; ɛ}}} & (2)\end{matrix}$modified for the acoustic (and electromagnetic) case by themultiplicative factor, k². This factor arises because, unlike for thequantum scattering case, the scattering interaction is of the form k²VThis extra factor, k², causes important changes in the scatteringbehavior compared to the quantal case as is elaborated in the AppendixA. Here k² is essentially the square of the spatial wavenumberparameter, H₀ describes the unperturbed wave propagation. Also, weexplicitly indicate that the abstract operators T_(k) and

G₀⁺depend on k as a parameter. The scattering amplitude for the processk→k′ then is given (using Dirac notation for compactness)

$\begin{matrix}{\left\langle {k^{\prime}{T_{k}}k} \right\rangle = {\left\langle {k^{\prime}{V}k} \right\rangle + \left\langle {k^{\prime}{{{VG}_{ok}^{+}T_{k}}}k} \right\rangle}} & (3)\end{matrix}$

Taking advantage of the local character of V, we then have that

$\begin{matrix}{\left\langle {k^{\prime}{T_{k}}k} \right\rangle = {\frac{1}{2\pi}{\int_{- \infty}^{+ \infty}\mspace{7mu}{{\mathbb{d}z}\;{\mathbb{e}}^{{{\mathbb{i}}{({k - k^{\prime}})}}z}{V(z)}}}}} & (4)\end{matrix}$

It is important to note that the matrix element <k′|V|k> of the truelocal inter-action only depends on the difference (k−k′). Conversely, if<k′|V|k> is only a function of (k−k′) then V is local. If V(z) is real,then we also have that<k′|V|k>=<k|V|k′>*  (5)It can then be established that <k′|V|k> and <k′|V|k′> cannotsimultaneously depend only on the difference (k−k) for if that were thecase, then Equation (1) could be written as

$\begin{matrix}\begin{matrix}{\left\langle {k^{\prime}{T_{k}}k} \right\rangle = {\left\langle {k^{\prime}{V}k} \right\rangle + {\int{{\mathbb{d}k^{''}}{\overset{\sim}{V}\left( {k^{''} - k^{\prime}} \right)}\left\langle {k^{''}{G_{0k}^{+}}k^{''}} \right\rangle{{\overset{\sim}{T}}_{k}\left( {k - k^{''}} \right)}}}}} \\{= {\left\langle {k^{\prime}{V}k} \right\rangle + {\int{{\mathbb{d}y}{\overset{\sim}{V}\left( {k - k^{\prime} - y} \right)}\left\langle {k^{''}{G_{0k}^{+}}k^{''}} \right\rangle{{\overset{\sim}{T}}_{k}(y)}}}}} \\{= {\left\langle {k^{\prime}{V}k} \right\rangle + {\int{{\mathbb{d}y}\;{\overset{\sim}{V}\left( {k - k^{\prime} - y^{\prime}} \right)}\frac{k^{2}}{{\left( {{2k} - y} \right)y} + {i\; ɛ}}{{\overset{\sim}{T}}_{k}(y)}}}}}\end{matrix} & (6)\end{matrix}$

The last line, which results from substitution of the Green's functionof Equation (2) into the previous line, clearly shows that the integral,and hence the LHS of this equation, is not solely a function of k−k′,which is a contradiction. For inversion, we write Equation (1) as anequation for <k′|V|k>

$\begin{matrix}{\left\langle {k^{\prime}{V}k} \right\rangle = {\left\langle {k^{\prime}{T_{k}}k} \right\rangle - \left\langle {k{{{VG}_{0k}^{+}T_{k}}}k} \right\rangle}} & (7)\end{matrix}$Using the resolution of the identity1=∫dk″|k″><k″|  (8)this equation becomes

$\begin{matrix}{\left\langle {k^{\prime}{V}k} \right\rangle = {\left\langle {k^{\prime}{T_{k}}k} \right\rangle - {\int{{\mathbb{d}k^{''}}\frac{\left\langle {k^{\prime}{V}k^{''}} \right\rangle\left\langle {k^{''}{T_{k}}k} \right\rangle}{k^{2} - k^{''2} + {i\; ɛ}}}}}} & (9)\end{matrix}$which is an exact equation. We first observe that if we take |k′|=|k| sothat <k′|T_(k)|k> corresponds to an on-shell matrix element, then<k′|V|k″> still involves both on-shell and half-on-shell T matrixelements.

As we have seen, although V is taken to be a local operator, it cannotbe true in general that T_(k) is local (or, more precisely, <k′|T_(k)|k>is not solely a function of k−k′). The approach of Moses, Razavy, andProsser [3A–5A] involves additional expansions of (k′Vk) and (k′T_(k)k)in a power series of the on-shell (reflection) amplitude. Thus, inaddition to the issue of convergence of the Born expansion of Equation(1), there is also the question of convergence of these expansionssolely in terms of the far field reflection amplitude. We note that thisanalysis also applies to the Volterra inverse integral equation. Onecan, in general, write the Volterra Green's operator, {tilde over(G)}_(0k), as

G_(0k)⁺plus a solution of the homogeneous free Green's function [7A, 8A]equation Such homogeneous solutions can always be written as a sum ofseparable, totally on-shell operators having the form

$\begin{matrix}{O_{k} = {\sum\limits_{n}^{\;}\;{\left. \phi_{nk} \right\rangle\left\langle \chi_{nk} \right.}}} & (10)\end{matrix}$where(k ² −H ₀)|φ_(nk)>=(k ² −H ₀)|χ_(nk)>=0  (11)Then,

$\begin{matrix}{G_{0k}^{+} = {{\overset{\sim}{G}}_{0k} + O_{k}}} & (12)\end{matrix}$(12)and,T _(k) =V+V{tilde over (G)} _(0k) T _(k) +VO _(k) T _(k) =V[1+O _(k) T_(k) ]+V{tilde over (G)} _(0k) T _(k)  (13)

We define {tilde over (T)}_(k) by{tilde over (T)} _(k) =V+V{tilde over (G)} _(0k) {tilde over (T)}_(k)  (14)where,T _(k) ={tilde over (T)} _(k)(1+O _(k) T _(k))  (15)

Again one can show that {tilde over (T)}_(k) is, in general, non-local,and a parallel analysis to that for T_(k) holds. The major distinctionbetween using Equation (3) to generate an inverse series for <k′|V|k>and using similar matrix elements of Equation (14), expressed asV={tilde over (T)} _(k) −V{tilde over (G)} _(0k) {tilde over (T)}_(k)  (16)is that the iterative solution of Equation (14) for V is guaranteed toconverge absolutely and uniformly no matter how strong V is [7A, 8A].There remains the question of whether the general matrix elements of Vand T_(k) can be expanded in convergent series of far-field amplitudes.Thus, whether one bases an inversion on T_(k) or {tilde over (T)}_(k),both require the equivalent of half-on-shell information for theirimplementation. We next examine the behavior of these two inversionalternatives with regard to their relations to moments of the trueinteraction, because this shows an important distinction in howfar-field quantities affect these two.III. Analysis of Moments of the Interaction

The moments of the interaction are defined as,

$\begin{matrix}{{V\lbrack n\rbrack} \equiv {\int_{- \infty}^{+ \infty}\ {{\mathbb{d}{zz}^{n}}{V(z)}}}} & (17)\end{matrix}$

It is clear from equations (7) and (16) that V(n) is exactly given by

$\begin{matrix}{{V\lbrack n\rbrack} = {\int_{- \infty}^{+ \infty}\ {{\mathbb{d}{zz}^{n}}{\int_{- \infty}^{+ \infty}\ {{\mathbb{d}k}\;{\mathbb{e}}^{{- 2}\;{\mathbb{i}}\; k\; z}\left\langle {{- k}{V}k} \right\rangle}}}}} & (18)\end{matrix}$leading to

$\begin{matrix}{{V\lbrack n\rbrack} = {\int_{- \infty}^{+ \infty}\ {{\mathbb{d}{zz}^{n}}{\int_{- \infty}^{+ \infty}\ {{\mathbb{d}k}\;{{\mathbb{e}}^{{- 2}\;{\mathbb{i}}\; k\; z}\left\lbrack {\left\langle {{- k}{T_{k}}k} \right\rangle - \left\langle {- k} \middle| {{VG}_{0k}^{+}T_{k}} \middle| k \right\rangle} \right\rbrack}}}}}} & (19) \\{or} & \; \\{{V\lbrack n\rbrack} = {\int_{- \infty}^{+ \infty}\ {{\mathbb{d}{zz}^{n}}{\int_{- \infty}^{+ \infty}\ {{\mathbb{d}k}\;{{\mathbb{e}}^{{- 2}\;{\mathbb{i}}\; k\; z}\left\lbrack {\left\langle {{- k}{{\overset{\sim}{T}}_{k}}k} \right\rangle - \left\langle {- k} \middle| {V\;{\overset{\sim}{G}}_{0k}{\overset{\sim}{T}}_{k}} \middle| k \right\rangle} \right\rbrack}}}}}} & (20)\end{matrix}$

It is generally assumed that <−k|T_(k)k> and <−k|{tilde over (T)}_(k)|k>are obtained experimentally by a far-field measurement. The termsinvolving

⟨−kVG_(0k)⁺T_(k)k⟩and the analogous matrix element of T_(k) are the source of far-fieldeffects in the above equations. Again, defining,

$\begin{matrix}{{V_{1}(z)} \equiv {\int_{- \infty}^{+ \infty}\ {{\mathbb{d}k}\;{\mathbb{e}}^{{- 2}\;{\mathbb{i}}\;{kz}}\left\langle {{- k}{T_{k}}k} \right\rangle}}} & (21) \\{{and},} & \; \\{{{\overset{\sim}{V}}_{1}(z)} \equiv {\int_{- \infty}^{- \infty}\ {{\mathbb{d}k}\;{\mathbb{e}}^{{- 2}\;{\mathbb{i}}\;{kz}}\left\langle {{- k}{{\overset{\sim}{T}}_{k}}k} \right\rangle}}} & (22) \\{{{we}\mspace{14mu}{have}},} & \; \\{{V\lbrack n\rbrack} = {{V_{1}\lbrack n\rbrack} - {\int_{- \infty}^{+ \infty}\ {{\mathbb{d}{zz}^{n}}{\int_{- \infty}^{+ \infty}\ {{\mathbb{d}k}\;{\mathbb{e}}^{{- 2}\;{\mathbb{i}}\;{kz}}\left\langle {{- k}{{{VG}_{0k}^{+}T_{k}}}k} \right\rangle}}}}}} & (23) \\{{and},} & \; \\{{V\lbrack n\rbrack} = {{{\overset{\sim}{V}}_{1}\lbrack n\rbrack} - {\int_{- \infty}^{= \infty}\ {{\mathbb{d}{zz}^{n}}{\int_{- \infty}^{+ \infty}\ {{\mathbb{d}k}\;{\mathbb{e}}^{{- 2}\;{\mathbb{i}}\;{kz}}\left\langle {{- k}{{V\;{\overset{\sim}{G}}_{0k}{\overset{\sim}{T}}_{k}}}k} \right\rangle}}}}}} & (24)\end{matrix}$

We shall now prove that, in general, {tilde over (V)}₁[n] is exact(i.e., {tilde over (V)}[n]={tilde over (V)}₁[n] through n =1 while V₁[n]is only correct for n=0. To do this, we substitute into these equationsthe coordinate representation matrix elements of

G_(0k)⁺and {tilde over (G)}_(0k), which are given respectively by [8A]

$\begin{matrix}{{G_{0k}^{+}\left( z \middle| z^{\prime} \right)} = {\frac{{- i}\; k}{2}{\mathbb{e}}^{{\mathbb{i}}\; k{{z^{\prime} - z}}}}} & (25)\end{matrix}$and{tilde over (G)} _(0k)(z|z′)= k sin(k[z′−z])H(z′−z)  (26)Here H(z′−z) is the Heaviside function,

$\begin{matrix}{{H(z)} = \left\{ \begin{matrix}{1,} & {z > 0} \\{0,} & {z \leq 0}\end{matrix} \right.} & (27)\end{matrix}$

After judicious insertion of identity resolutions, we then obtain theresults

$\begin{matrix}{{V\lbrack n\rbrack} = {{V_{1}\lbrack n\rbrack} - {\frac{i}{2}{\int_{\infty}^{+ \infty}\mspace{7mu}{{\mathbb{d}{zz}^{n}}{\int_{\infty}^{+ \infty}\mspace{7mu}{{\mathbb{d}k}{\int_{\infty}^{+ \infty}{{\mathbb{d}z^{\prime}}{\int_{\infty}^{+ \infty}{{\mathbb{d}z^{''}}{\mathbb{e}}^{{- 2}{\mathbb{i}}\;{kz}}{\mathbb{e}}^{{\mathbb{i}kz}^{\prime}}{V\left( z^{\prime} \right)}k\;{\mathbb{e}}^{{\mathbb{i}}\; k{{z^{\prime} - z^{''}}}}\left\langle {z^{''}{T_{k}}k} \right\rangle}}}}}}}}}}} & (28)\end{matrix}$and

$\begin{matrix}{{V\lbrack n\rbrack} = {{{\overset{\sim}{V}}_{1}\lbrack n\rbrack} - {\int_{\infty}^{+ \infty}\mspace{7mu}{{\mathbb{d}{zz}^{n}}{\int_{\infty}^{+ \infty}\mspace{7mu}{{\mathbb{d}k}{\int_{\infty}^{+ \infty}{{\mathbb{d}z^{\prime}}{\int_{\infty}^{+ \infty}{{\mathbb{d}z^{''}}{\mathbb{e}}^{{- 2}{\mathbb{i}kz}}{\mathbb{e}}^{{\mathbb{i}}\;{kz}^{\prime}}{V\left( z^{\prime} \right)}k\;{\sin\left( {k\left\lbrack {z^{''} - z^{\prime}} \right\rbrack} \right)}{H\left( {z^{''} - z^{\prime}} \right)}\left\langle {z^{''}{{\overset{\sim}{T}}_{k}}k} \right\rangle}}}}}}}}}} & (29)\end{matrix}$

Next we interchange the order of the dz and dk integrals and note that

$\begin{matrix}{{\int_{\infty}^{+ \infty}\mspace{7mu}{{\mathbb{d}{zz}^{n}}{\mathbb{e}}^{{- 2}{ikz}}}} = {{\left( \frac{i}{2} \right)^{n}\frac{\partial^{n}}{\partial k^{n}}{\int_{\infty}^{+ \infty}\ {{\mathbb{d}z}\;{\mathbb{e}}^{{- 2}{ikz}}}}} = {2{\pi\left( \frac{i}{2} \right)}^{n}\frac{\partial^{n}}{\partial k^{n}}{\delta\left( {2k} \right)}}}} & (30)\end{matrix}$

Consequently, Equations (28) and (29) become

$\begin{matrix}{{V\lbrack n\rbrack} = {{V_{1}\lbrack n\rbrack} + {\frac{\pi\; i^{n}}{2^{n - 1}}{\int_{\infty}^{+ \infty}\mspace{7mu}{{\mathbb{d}z^{\prime}}{\int_{\infty}^{+ \infty}\mspace{7mu}{{\mathbb{d}z^{''}}{V\left( z^{\prime} \right)}\frac{\partial^{n}}{\partial k^{n}}\left\{ {{\mathbb{e}}^{{\mathbb{i}}\;{ikz}^{\prime}}k\;{\mathbb{e}}^{{\mathbb{i}}\; k{{z^{\prime} - z^{''}}}}\left\langle {z^{''}{T_{k}}k} \right\rangle} \right\}_{k = 0}}}}}}}} & (31)\end{matrix}$and

$\begin{matrix}{{V\lbrack n\rbrack} = {{{\overset{\sim}{V}}_{1}\lbrack n\rbrack} + {\frac{\pi\; i^{n + 1}}{2^{n}}{\int_{\infty}^{+ \infty}\mspace{7mu}{{\mathbb{d}z^{\prime}}{\int_{\infty}^{+ \infty}\mspace{7mu}{{\mathbb{d}z^{''}}{V\left( z^{\prime} \right)}{H\left( {z^{''} - z^{\prime}} \right)}\frac{\partial^{n}}{\partial k^{n}}\left\{ {{\mathbb{e}}^{{\mathbb{i}}\;{ikz}^{\prime}}k\;{\sin\left\lbrack {k\left( {z^{''} - z^{\prime}} \right)} \right\rbrack}\left\langle {z^{''}{T_{k}}k} \right\rangle} \right\}_{k = 0}}}}}}}} & (32)\end{matrix}$

Now the essential point to note is that when n=0 the second terms inboth Equations (31) and (32) vanish since there is no derivative andk=0. Therefore, we conclude thatV[0]=V ₁[0]={tilde over (V)} ₁[0]  (33)However, when n=1, there is a nonzero contribution from the second termon the RHS of Equation (31) since one term contains

${\frac{\partial\;}{\partial k}(k)} = 1$and the remaining factors are non-zero, in general. The second term onthe RHS in Equation (32) remains zero since

$\begin{matrix}{{\frac{\partial\;}{\partial k}\left\lbrack {k\mspace{14mu}{\sin\left\lbrack {k\left( {z^{''} - z^{\prime}} \right)} \right\rbrack}} \right\rbrack}_{k = 0} = 0} & (34)\end{matrix}$and we thus conclude thatV[1]={tilde over (V)}[1]  (35)

Thus, the approximation to the interaction produced by the Volterraformalism yields correct values for V[0] and V[1] while theFredholm-based Born series yields the correct values only for V[0]. Inanother portion of this application, we will present a completeinversion scheme based on moment of V using the iterated Volterra-Bornseries [9A]. We comment that the first order Volterra approximation,{tilde over (V)}₁ is not as sensitive to the near-field effects as thefirst order Fredholm approximation, V₁.

We now turn to consider the Volterra-based inversion for interactionsthat can be expressed as sums of Dirac delta functions.

IV. Volterra-Based Inverse Scattering Treatment for Sums of Dirac DeltaFunctions

Rodberg and Thaler [10] have presented an interesting derivation ofFredholm's method for solving the Lippmann-Schwinger equation thatinvolves writing the interaction as a sum of Dirac delta functions.Essentially they argue that since

$\begin{matrix}{{V(z)} = {\int_{\infty}^{+ \infty}\mspace{7mu}{{\mathbb{d}z^{\prime}}{\delta\left( {z^{\prime} - z} \right)}{V\left( z^{\prime} \right)}}}} & (36)\end{matrix}$holds for reasonable interaction functions, one can write V(z) as alimit

$\begin{matrix}{{V(z)} = {\lim\limits_{{\Delta\; j}->0}{\sum\limits_{j}^{\;}\;{\Delta_{j}{\delta\left( {z_{j} - z} \right)}{V\left( z_{j} \right)}}}}} & (37)\end{matrix}$

While care must be exercised with such an argument (as shown in moredetail in Appendix B), it suggests that a useful model, especially for ascattering interaction having effective compact support and which isalso effectively band-limited (of course, both are not rigorouslypossible simultaneously), can be taken to be

$\begin{matrix}{{V(z)} = {\sum\limits_{j = 1}^{J}\;{\Delta_{j}{\delta\left( {z_{j} - z} \right)}{V\left( z_{j} \right)}}}} & (38)\end{matrix}$where J is the finite number of delta functions needed to represent theinter-action adequately. (By using Hermnite distributed approximatingfunctionals (HDAFs), δ_(M)(z−z_(j)|jσ)) to replace the delta functions,we can obtain the smooth, well-behaved interaction form

$\begin{matrix}{{V(z)} = {\sum\limits_{j}^{\;}{\delta_{M}\left( {z - {z_{j}\left. \sigma \right){V\left( z_{j} \right)}}} \right.}}} & (39)\end{matrix}$about which we later make some brief comments in Appendix B. In thissection of the application, we shall focus primarily on Equation (38).)

We recall that the Volterra-based solution of the 1−D scalar Helmholtzequation is [7]

$\begin{matrix}{{{\overset{\sim}{\psi}}_{k}(z)} = {\frac{{\mathbb{e}}^{{\mathbb{i}}\;{ka}}}{2\pi} + {k{\int_{- \infty}^{+ \infty}{{\mathbb{d}z^{\prime}}{\sin\left\lbrack {k\left( {z^{\prime} - z} \right)} \right\rbrack}{H\left( {z^{\prime} - z} \right)}{V\left( z^{\prime} \right)}{\overset{\sim}{\psi}\left( z^{\prime} \right)}}}}}} & (40)\end{matrix}$

For the interaction Equation (38), this is seen to give

$\begin{matrix}{{{\overset{\sim}{\psi}}_{k}(z)} = {\frac{{\mathbb{e}}^{{\mathbb{i}}\;{ka}}}{2\pi} + {k{\sum\limits_{j = 1}^{J}{{\sin\left\lbrack {k\left( {z_{j} - z} \right)} \right\rbrack}{H\left( {z_{j} - z} \right)}{\Delta\;}_{j}{V\left( z_{j} \right)}{\overset{\sim}{\psi}\left( z_{j} \right)}}}}}} & (41)\end{matrix}$

We assume, without loss of generality, thatz_(j)>z_(j−1)  (42)In terms of the full Green's operator, we have that{tilde over (ψ)}_(k)(z)=<z|k >+<z|{tilde over (G)}V|k)  (43)which, making use of the well known expansion

$\begin{matrix}{G = {\sum\limits_{j = 0}^{\;}{\left( {{\overset{\sim}{G}}_{0}V} \right)^{j}{\overset{\sim}{G}}_{0}}}} & (44)\end{matrix}$give rise to

$\begin{matrix}{{{\overset{\sim}{\psi}}_{k}(z)} = \left\langle {{z\left. k \right\rangle} + {\sum\limits_{n = 1}^{\;}\left\langle {z{{{\left( {{\overset{\sim}{G}}_{0}V} \right)^{n}\left. k \right\rangle} = \left\langle {{z\left. k \right\rangle} + {\sum\limits_{n = 1}^{\;}{\int_{z < \varsigma_{1}}^{\infty}{{\mathbb{d}\varsigma_{1}}{\int_{\varsigma_{1} < \varsigma_{2}}^{\infty}{{\mathbb{d}\varsigma_{2}}\ldots{\int_{\varsigma_{n - 1} < \varsigma_{n}}^{\infty}{{\mathbb{d}\varsigma_{n}}\left\langle {z{{{\overset{\sim}{G}}_{0}V}}\varsigma_{1}} \right\rangle\left\langle {\varsigma_{1}{{{\overset{\sim}{G}}_{0}V}}\varsigma_{2}} \right\rangle\ldots\left\langle {\varsigma_{n - 1}{{{\overset{\sim}{G}}_{0}V}}\varsigma_{n}} \right\rangle\left\langle {\varsigma_{n}\left. k \right\rangle} \right.}}}}}}}} \right.}}} \right.}} \right.} & (45)\end{matrix}$

Here, for clarity we have used ζ instead of z as the symbol to representthe ordered coordinate integration variables. This formula is general,but if we now introduce the interaction of Equation (38) the integralscan be evaluated to yield

$\begin{matrix}{{{\overset{\sim}{\psi}}_{k}(z)} = \left\langle {{z\left. k \right\rangle} + {\sum\limits_{n = 1}^{\;}{\sum\limits_{z < \varsigma_{1}}^{z_{J}}{\sum\limits_{\varsigma_{1} < \varsigma_{2}}^{z_{J}}{\ldots{\sum\limits_{\varsigma_{n - 1} < \varsigma_{n}}^{z_{J}}{\left\langle {z{{{\overset{\sim}{G}}_{0}V}}\varsigma_{1}} \right\rangle\left\langle {\varsigma_{1}{{{\overset{\sim}{G}}_{0}V}}\varsigma_{2}} \right\rangle\ldots\left\langle {\varsigma_{n - 1}{{{\overset{\sim}{G}}_{0}V}}\varsigma_{n}} \right\rangle\left\langle {\varsigma_{n}\left. k \right\rangle} \right.}}}}}}} \right.} & (46)\end{matrix}$where ζ₁ through ζ_(n) are now elements of an ordered subset of thez-points where the &functions of the interaction are located. LetN_(z)≦J be the number of such points greater than z, then the number ofordered sets of ζ-points satisfying the limits on the summations isgiven by the binomial coefficient N_(z)!/[n!(N_(z)−n)!]. The sum

${\sum\limits_{z < \varsigma_{1}}^{z_{J}}{\sum\limits_{\varsigma_{1} < \varsigma_{2}}^{z_{J}}{\ldots\sum\limits_{\varsigma_{n - 1} < \varsigma_{n}}^{z_{J}}}}},$which for conciseness we write as Σ_(ζ) ₁ _(,ζ) ₂ _(, . . . ζ) _(n) isthe sum over all N_(z)!/[n!(N_(z)−n)!] sets. Finally, we have that

$\begin{matrix}{{{\overset{\sim}{\psi}}_{k}(z)} = \left\langle {{z\left. k \right\rangle} + {\sum\limits_{n = 1}^{N_{z}}{\sum\limits_{\varsigma_{1},\varsigma_{2},{\ldots\varsigma}_{n}}^{\;}{\left\langle {z{{{\overset{\sim}{G}}_{0}V}}\varsigma_{1}} \right\rangle\left\langle {\varsigma_{1}{{{\overset{\sim}{G}}_{0}V}}\varsigma_{2}} \right\rangle\ldots\left\langle {\varsigma_{n - 1}{{{\overset{\sim}{G}}_{0}V}}\varsigma_{n}} \right\rangle\left\langle {\varsigma_{n}\left. k \right\rangle} \right.}}}} \right.} & \left( {47a} \right) \\{{{\overset{\sim}{\psi}}_{k}(z)} = \left\langle {{z\left. k \right\rangle} + {\sum\limits_{n = 1}^{N_{z}}{\left( {k\;\Delta} \right)^{n}{\sum\limits_{\varsigma_{1},\varsigma_{2},{\ldots\varsigma}_{n}}^{\;}{{\sin\left( {k\left\lbrack {\varsigma_{1} - z} \right\rbrack} \right)}{V\left( \varsigma_{1} \right)}{\sin\left( {k\left\lbrack {\varsigma_{2} - \varsigma_{1}} \right\rbrack} \right)}{V\left( \varsigma_{2} \right)}\ldots\;{\sin\left( {k\left\lbrack {\varsigma_{n} - \varsigma_{n - 1}} \right\rbrack} \right)}{V\left( \varsigma_{n} \right)}\left\langle {\varsigma_{n}\left. k \right\rangle} \right.}}}}} \right.} & \left( {47b} \right)\end{matrix}$

It is clear that the number of terms contributing to {tilde over(ψ)}_(k) (z), for any value of z, is

${{1 + {\sum\limits_{n = 1}^{N_{z}}\frac{N_{z}!}{{n!}{\left( {N_{z} - n} \right)!}}}} = 2^{N_{z}}},$which is finite (again assuming the number of delta functions in theinteraction to be finite). Thus, for z≧z_(J), N_(z)=0 and only one termcontributes to {tilde over (ψ)}_(k)(z). That is{tilde over (ψ)}_(k)(z)=<z|k>  (48)

Similarly, for z_(J)>z≧z_(j−1), N_(z)=1 and we have{tilde over (ψ)}_(k)(z)=<z|k>+kΔ _(J) sin(k[z _(J) −z])V(z _(J)) <z_(J|k>)  (49)and for z_(J−1)>z≧z_(J−2), N_(z)=2 and

$\begin{matrix}\begin{matrix}{{{\overset{\sim}{\psi}}_{k}(z)} = \left\langle {{z\left. k \right\rangle} +} \right.} \\{k\;\Delta_{J}{\sin\left( {k\left\lbrack {z_{J} - z} \right\rbrack} \right)}{V\left( z_{J} \right)}\left\langle {{z_{J}\left. k \right\rangle} +} \right.} \\{k\;\Delta_{J - 1}{\sin\left( {k\left\lbrack {z_{J - 1} - z} \right\rbrack} \right)}{V\left( z_{J - 1} \right)}\left\langle {{z_{J - 1}\left. k \right\rangle} +} \right.} \\{k^{2}\Delta_{J - 1}\Delta_{J}{\sin\left( {k\left\lbrack {z_{J - 1} - z} \right\rbrack} \right)}{V\left( z_{J - 1} \right)}{\sin\left( {k\left\lbrack {z_{J} - z} \right\rbrack} \right)}{V\left( z_{J} \right)}\left\langle {z_{J}{k}} \right\rangle}\end{matrix} & (50)\end{matrix}$etc. As one progresses in this manner from the transmission to thereflection region, the number of terms proliferate, but they areindividually quite simple. Finally, in the reflection region the numberof terms in the wave function is two raised to the number of scatteringpoints in the interaction. Obviously, if we had such a progression of{tilde over (ψ)}_(k) values it would be trivial to solve for the variousV(z_(j)) values sequentially starting from the transmission end.

It is instructive to express this result using an N_(z)+1 dimensionalvector representation where one component stands for the point z and theother N_(z) components represent the delta function points in theinteraction on the trans-mission side of z. To this end we introduce aset of N_(z) +1 orthogonal unit vectors |0}, |1}, |2} . . . |Nz}, where|0} is the unit vector associated with z. Here we use Dirac notationwith |j} representing the j^(th) unit vector in this finite dimensionalspace. Then we have that {s|t}=δ_(s,t) and the identity matrix in thisspace is given by

$\begin{matrix}{1 = {\sum\limits_{s = 0}^{N_{z}}{\left. s \right\}\left\{ s \right.}}} & (51)\end{matrix}$

We next define the matrix Y by{j|Y|l}=sin(k[z _(l) −z _(j)])V(z _(l))Δ_(l)  (52)Then

$\begin{matrix}{{{\overset{\sim}{\psi}}_{k}(z)} = \left\{ {0\left. {\left\lbrack {1 + {kY}} \right.1} \right\}\left\{ {1{\left.  \right\rbrack\left\lbrack {1 + {{kY}\left. 2 \right\}\left\{ {2\left.  \right\rbrack\mspace{11mu}{\ldots\mspace{11mu}\left\lbrack {1 + {{kY}\left. N_{z} \right\}\left\{ {N_{z}\left.  \right\rbrack} \right.l}} \right\}}{\sum\limits_{l = 0}^{N_{z}}{\left. l \right\}\left\langle {z_{l}\left. k \right\rangle} \right.}}} \right.}} \right.}} \right.} \right.} & (53)\end{matrix}$which provides an explicit summation of the Volterra-Born series forthis interaction. Physically, we see that each scattering center (i.e.,delta function) in the interaction either produces a reflection or ithas no effect. Hence the wavefunction at any point is only aware of thescattering centers that lie to the transmission side. The Volterra-Bornseries at a point z is simply a finite sum of the 2^(N)= possibilities.Finally, if we knew {tilde over (ψ)}_(k)(z) in the reflection region atas many values of k as there are scattering points in the interaction wecould in principle solve the resulting (highly non-linear) equations forthe V(z_(j)). We stress that such a procedure corresponds to usingfar-field measurements to determine near-field quantities exactly(essentially, one is obtaining the {tilde over (ψ)}_(k)(z_(j))).

One could, of course, follow exactly the same procedure in the Fredholmcase starting with

$\begin{matrix}{{{\overset{\sim}{\psi}}_{k}^{+}(z)} = \left\langle {{z\left. k \right\rangle} + \left\langle {z{{G^{+}V}}k} \right\rangle} \right.} & (54)\end{matrix}$However, in this case there is no Heaviside-function in the coordinaterepresentation of the Green's operator and consequently the counterpartto Equation (46) does not have an ordered sum. The result is that thesum of terms contributing to

ψ_(k)⁺(z)for any value of z is infinite. This simple interaction illustrates thecomparative convergence properties of the Fredholm-Born andVolterra-Born series.

We next note that the {tilde over (T)}-matrix element

$\begin{matrix}{\left\langle {{- k}{{\overset{\sim}{T}}_{k}}k} \right\rangle = {\left\langle {{- k}{V}{\overset{\sim}{\psi}}_{k}} \right\rangle = {\sum\limits_{j = 1}^{J}{{\mathbb{e}}^{{\mathbb{i}}\;{kz}_{j}}{V\left( z_{j} \right)}\Delta_{j}{{\overset{\sim}{\psi}}_{z}\left( z_{j} \right)}}}}} & (55)\end{matrix}$From Equation (22), we have that

$\begin{matrix}{{V_{1}(z)} = {\int_{- \infty}^{- \infty}{{\mathbb{d}k}\;{\mathbb{e}}^{{- 2}\;{\mathbb{i}}\;{kz}}{\sum\limits_{j = 1}^{J}{{\mathbb{e}}^{{\mathbb{i}}\;{kz}_{j}}{V\left( z_{j} \right)}\Delta_{j}{{\overset{\sim}{\psi}}_{z}\left( z_{j} \right)}}}}}} & (56)\end{matrix}$and thus, from Equation (50), we see that the general form of {tildeover (ψ)}_(k)(z_(j)) is{tilde over (ψ)}_(k)(z _(j))=<z_(j) |k >+R(k)  (57)where the various terms in R(k) have two kinds of k dependence. First,each is proportional to a linear or higher power of k and, second, eachcontains phase factor exponentials (resulting from decomposition of thesine functions). Hence, the k-integral of each term can be evaluatedexplicitly. The integral of the (z_(j)|k) terms simply reproduce theoriginal interaction (corresponding to the first order Bornapproximation), and, since

$\begin{matrix}{{k^{l}{\mathbb{e}}^{{- 2}\;{\mathbb{i}}\;{kz}}} = {\left( \frac{i}{2} \right)^{l}\frac{\partial^{l}\;}{\partial z^{l}}{\mathbb{e}}^{{- 2}\;{\mathbb{i}}\;{kz}}}} & (58)\end{matrix}$the terms arising from R(k) contain first or higher derivatives (withrespect to z) of delta functions.

For example, the special case of J=3 is

$\begin{matrix}{{V_{1}(z)} = {{\sum\limits_{j = 1}^{3}{\Delta_{j}{V\left( z_{j} \right)}{\delta\left( {z - z_{j}} \right)}}} + {\frac{1}{4}{\sum\limits_{j = 1}^{2}{\sum\limits_{j^{\prime} > j}^{\;}{\Delta_{j}\Delta_{j^{\prime}}{V\left( z_{j} \right)}{{V\left( z_{j^{\prime}} \right)}\left\lbrack {{\delta^{\prime}\left( {z_{j^{\prime}} - z} \right)} - {\delta^{\prime}\left( {z_{j} - z} \right)}} \right\rbrack}}}}} + {\frac{\Delta_{1}\Delta_{2}\Delta_{3}}{16}\left\lbrack \left. \quad{{\delta^{''}\left( {z_{3} - z} \right)} - {\delta^{''}\left( {z_{2} - z} \right)} + {\delta^{''}\left( {z_{1} - z} \right)} - {\delta^{''}\left( {z_{1} + z_{3} - z_{2} - z} \right)}} \right\rbrack \right.}}} & (59)\end{matrix}$If there are more sampling points in the interaction, the structureremains analogous but there occur higher order derivatives of the Diracdelta functions. Finally, we note that formally (and exactly)

$\begin{matrix}{{{\int_{z_{j} - {\Delta_{j}/2}}^{z_{j} + {\Delta_{j}/2}}{{\mathbb{d}z}\;{\delta^{p}\left( {z - z_{j}} \right)}}} = 0},{p \geq 1}} & (60)\end{matrix}$

It follows that averaging V₁ in the neighborhood of a sampling pointaverages all of the higher terms (i.e., the near-field or half-off-shelleffects) to zero and we find that

$\begin{matrix}{{\int_{z_{j} - {\Delta_{j}/2}}^{z_{j} + {\Delta_{j}/2}}{\mathbb{d}{{zV}_{1}(z)}}} = {\Delta_{j}{V\left( z_{j} \right)}}} & (61)\end{matrix}$Once the V(z_(j)) are known, one knows the interaction.

Of course, such averaging is a formal exercise for a interaction that isa sum of delta functions since both V(z) and V₁(z) are not truefunctions. They only have meaning in terms of delta sequences. However,the operational procedure can be applied to real data to construct anHDAF-approximation where the HDAF is interpreted as a member of a deltasequence. Thus, the suggested procedure would be to take the on-shell(far-field) amplitudes <−k|{tilde over (T)}_(k)|k> and evaluate {tildeover (V)}₁(z) by Equation (22). Then one would use some averagingprocedure such as that indicated in Equation (61) to obtain approximateexpressions for the Δ_(j)V(z_(j)) on a sufficiently dense set of pointsto construct an acceptable approximation to the true interaction usingEquation (39). In the process of this averaging, one is taking accountof the effects of near-field terms in the Volterra integral equation.

V. Implementation of the Volterra-Based Inversion

As is evident from Equation (61), if we have the modified reflectioncoefficient <−k|{tilde over (T)}_(k)|k>, we can evaluate the V(z_(j))parameters approximately by a suitable averaging procedure. However,experiments are generally carried out under conditions that do not makedirect measurement of <−k|{tilde over (T)}_(k)|k) possible. But, asshown previously, this quantity is calculated from the physicalreflection <−k|T|k> and transmission <k|T|k> amplitudes [7]. Thus toapply Equations (55), (56) and (61) immediately requires an additionalmeasurement compared to a Fredholm-based inversion. This is the priceone pays to obtain the simplified Volterra expressions. It issignificant, nevertheless, that even if one cannot measure thetransmission <k|T|k>, the Volterra-based inversion can still be carriedout, albeit with substantially greater required effort. This greatereffort is the price one must pay to take account of the near-fieldeffects in a direct fashion.

To see how this can be done, we consider the Lippmann-Schwinger equation

$\begin{matrix}{{\psi_{k}^{+}(z)} = {{\mathbb{e}}^{{\mathbb{i}}\;{kz}} - {\frac{{\mathbb{i}}\; k}{2}{\int_{- \infty}^{+ \infty}{{\mathbb{d}z^{\prime}}{\mathbb{e}}^{{\mathbb{i}}\; k{{z - z^{\prime}}}}{V\left( z^{\prime} \right)}{\psi_{k}^{+}\left( z^{\prime} \right)}}}}}} & (62)\end{matrix}$

For the interaction form of Eq.(38), this yields

$\begin{matrix}{{\psi_{k}^{+}(z)} = {{\mathbb{e}}^{{\mathbb{i}}\;{kz}} - {\frac{{\mathbb{i}}\; k\;\Delta}{2}{\sum\limits_{j}^{\;}{{\mathbb{e}}^{{\mathbb{i}}\; k{{z - z_{j}}}}{V\left( z_{j} \right)}{\psi_{k}^{+}\left( z_{j} \right)}}}}}} & (63)\end{matrix}$

We find the transmission coefficient

$\begin{matrix}\begin{matrix}{t_{k} = {1 - {\frac{{\mathbb{i}}\; k}{2}{\int_{- \infty}^{+ \infty}{{\mathbb{d}z}\;{\mathbb{e}}^{{\mathbb{i}}\;{kz}}{V(z)}{\psi_{k}^{+}(z)}}}}}} \\{= {1 - {\frac{{\mathbb{i}}\; k\;\Delta}{2}{\sum\limits_{j}^{\;}{{\mathbb{e}}^{{- {\mathbb{i}}}\;{kz}_{j}}{V\left( z_{j} \right)}{\psi_{k}^{+}\left( z_{j} \right)}}}}}}\end{matrix} & (64)\end{matrix}$

The reflection coefficient similarly is

$\begin{matrix}{r_{k} = {{- \frac{{\mathbb{i}}\; k\;\Delta}{2}}{\sum\limits_{j}^{\;}{{\mathbb{e}}^{{- {\mathbb{i}}}\;{kz}_{j}}{V\left( z_{j} \right)}{\psi_{k}^{+}\left( z_{j} \right)}}}}} & (65)\end{matrix}$

The Volterra normalization is such that {tilde over (t)}_(k)≡1. Toachieve this we note that

$\begin{matrix}{{t_{k}{{\overset{\sim}{\psi}}_{k}(z)}} = {\psi_{k}^{+}(z)}} & (66)\end{matrix}$

Then by Equation (64),

$\begin{matrix}{{t_{k} = \frac{1}{1 + {\frac{{\mathbb{i}}\; k\;\Delta}{2}{\sum\limits_{j}^{\;}{{\mathbb{e}}^{{- {\mathbb{i}}}\;{kz}_{j}}{V\left( z_{j} \right)}{{\overset{\sim}{\psi}}_{k}\left( z_{j} \right)}}}}}}\mspace{14mu}{and}} & (67) \\{{\frac{r_{k}}{t_{k}} = {\overset{\sim}{r}}_{k}}\mspace{14mu}{{Thus},}} & (68) \\{{{r_{k}\left\lbrack {1 + {\frac{{\mathbb{i}}\; k\;\Delta}{2}{\sum\limits_{j}^{\;}{{\mathbb{e}}^{{- {\mathbb{i}}}\;{kz}_{j}}{V\left( z_{j} \right)}{{\overset{\sim}{\psi}}_{k}\left( z_{j} \right)}}}}} \right\rbrack} = {\overset{\sim}{r}}_{k}}{and}} & (69) \\{{{{\overset{\sim}{V}}_{1}(z)} = {\int_{- \infty}^{+ \infty}{{\mathbb{d}\left( {2k} \right)}\frac{2{\mathbb{i}}}{k}{\overset{\sim}{r}}_{k}{\mathbb{e}}^{{- 2}\;{\mathbb{i}}\;{kz}}}}}{yielding}} & (70) \\{{{\overset{\sim}{V}}_{1}(z)} = {\int_{- \infty}^{+ \infty}{{\mathbb{d}\left( {2k} \right)}{\mathbb{e}}^{{- 2}\;{\mathbb{i}}\;{kz}}\frac{2{\mathbb{i}}}{k}{r_{k}\left\lbrack {1 + {\frac{{\mathbb{i}}\; k\;\Delta}{2}{\sum\limits_{j}^{\;}{{\mathbb{e}}^{{- {\mathbb{i}}}\;{kz}_{j}}{V\left( z_{j} \right)}{{\overset{\sim}{\psi}}_{k}(z)}}}}} \right\rbrack}}}} & (71)\end{matrix}$To use this expression, we substitute for {tilde over (ψ)}_(k)(z r)using Equation (53), and carry out the averages in Equation (61) togenerate a system of nonlinear algebraic equations for the {tilde over(φ)}_(k)(z_(j)) parameters. The only experimental data then needed arethe r_(k).VI. Conclusion

In this portion of the application, we have considered the problem oftaking account of half-on-shell matrix elements of either T_(k) or{tilde over (T)}_(k). We considered the moments of the interactionexpressed in terms of T_(k) and {tilde over (T)}_(k) and proved that{tilde over (V)}₁[n], n=0, 1 is exact, while only V₁[0] is exact. Thissuggests that an inversion based on the Volterra scheme is preferred,due to the different manner in which the half-off-shell effects enter.This is further supported by the superior convergence properties of theVolterra-based inverse scattering series. We illustrated theseconvergence properties using a simple model interaction that isexpressed as a sum of Dirac delta functions. It was shown that a formallocal average of {tilde over (V)}₁ in the neighborhood of a deltafunction sampling point yields exactly the desired sampling value,Δ_(j)V(z_(j)). This is, of course, not exactly true for a realinteraction. However, as argued by Rodberg and Thaler [10A], using asufficiently dense sampling, combined with an HDAF replacement of theDirac delta functions, will yield a reasonable approximation [11A]. Ithas the advantage that the averaging process is, in essence, takingaccount of the half-off-shell contributions since it averages themexactly to zero for any interaction constructed as a sum of Dirac deltafunctions. This delta function approach, strictly speaking, depends ontaking a specific form for the interaction, but its application can bevery general when the interact parameters are determined as discussed.

Appendix A

The Modification of

G_(0k)⁺by the Factor of k²

Recall that the acoustic and electromagnetic wave scattering (in 1−D) isof the form k²V The Lippmann-Schwinger equation is

$\begin{matrix}{\left. \psi_{k}^{+} \right\rangle = {\left. k \right\rangle + {{G_{0k}^{+}\left( {k^{2}V} \right)}\left. \psi_{k}^{+} \right\rangle}}} & (72)\end{matrix}$The transition operator is then defined as

$\begin{matrix}{{T_{k}\left. k \right\rangle} = {{V\left. \psi_{k}^{+} \right\rangle} = {\left( {V + {{VG}_{0k}^{+}k^{2}T_{k}}} \right)\left. k \right\rangle}}} & (73)\end{matrix}$It is then convenient to absorb the k²-factor into the Green's sfunction as is done in Equation (2). This k²-factor has a consequence ofmaking a Born-expansion solution for T_(k) a low-k approximation [6].This is in sharp contrast to quantum scattering for which it is wellknown that the Born series always converges for sufficiently large k[8A, 12A]. In the Volterra case, this k²-factor cannot preventconvergence no matter how large k is.Appendix B

Representation of V(z) using Dirac Delta Functions

In a series of studies [11A] it has been shown that a well behavedfunction can be represented to controllable accuracy by an HDAFapproximation. From this point of view, the model interaction form

$\begin{matrix}{{V(z)} = {\sum\limits_{j}^{\;}{\delta_{M}\left( {z - {z_{j}\left. \sigma \right){V\left( z_{j} \right)}}} \right.}}} & (74)\end{matrix}$(see Equation (35)), with a suitable choice of the imbedded parametersV(z_(j)), can be made to approximate any realistic interaction closely.(Of course, a limitless number of delta-function approximations could beutilized, but as we later discuss there are advantages to the HDAFapproximation.) If V is effectively compact (and band-limited) then onlya finite number of V(z_(i)) parameters are needed. The Born expansion ofthe wavefunction is given by

$\begin{matrix}{{\psi_{k}(z)} = \left\langle {{z\left. k \right\rangle} + {\sum\limits_{n = 1}^{\;}\left\langle {z{{\left( {G_{0}V} \right)^{n}\left. k \right\rangle}}} \right.}} \right.} & (75)\end{matrix}$where ψ_(k)(z) and G₀ can represent either

ψ_(k)⁺(z)and G₀ ⁺ or {tilde over (ψ)}_(k)(z) and {tilde over (G)}₀. Since V is alocal operator

$\begin{matrix}\left\langle {z{{{\left( {G_{0}V} \right)^{n}\left. k \right\rangle} = {\int_{- \infty}^{+ \infty}{{\mathbb{d}z^{''}}\left\langle {z{G_{0}}z^{''}} \right\rangle{V\left( z^{''} \right)}\left\langle {z^{''}{{\left( {G_{0}V} \right)^{n - 1}\left. k \right\rangle}}} \right.}}}}} \right. & (76)\end{matrix}$and hence for this model interaction we have

$\begin{matrix}\left\langle {z{{{\left( {G_{0}V} \right)^{n}\left. k \right\rangle} = {\sum\limits_{j}^{\;}\left\{ {\int_{- \infty}^{+ \infty}{{\mathbb{d}z^{''}}\left\langle {z{G_{0}}z^{''}} \right\rangle{\delta_{M}\left( {z^{''} - {z_{j}\left. \sigma \right)\left\langle {z^{''}\left. {\left( {G_{0}V} \right)^{n - 1}\left. k \right\rangle} \right\}{V\left( z_{j} \right)}} \right.}} \right.}}} \right.}}}} \right. & (77)\end{matrix}$

Now δ_(M)(z″−z_(j)|σ), as a member of a delta sequence, is highlylocalized around z_(j). In fact, the degree of localization can becontrolled by our choice of the width parameter, σ. The flexibility ofthe interaction form, V(z), of Equation (74) to represent any realisticinteraction depends on its being “well-tempered,” as has been discussedelsewhere [11]. This essentially requires that as ay is decreased thenumber of z_(j)-points must be increased. By suitably localizingδ_(M)(z″−z_(j)|σ) relative to how the rapidly the rest of the integrandvaries, we can write

$\begin{matrix}{\int_{- \infty}^{+ \infty}{{\mathbb{d}z^{''}}\left\langle {z{G_{0}}z^{''}} \right\rangle{\delta_{M}\left( {z^{''} - {z_{j}\left. \sigma \right)\left\langle {z^{''}{{{\left( {G_{0}V} \right)^{n - 1}\left. k \right\rangle} \cong {\int_{- \infty}^{+ \infty}{{\mathbb{d}z^{''}}\left\langle {z{G_{0}}z_{j}} \right\rangle{\delta_{M}\left( {z^{''} - {z_{j}\left. \sigma \right)\left\langle {z_{j}{{\left( {G_{0}V} \right)^{n - 1}\left. k \right\rangle}}} \right.}} \right.}}}}}} \right.}} \right.}}} & (78)\end{matrix}$to arbitrary accuracy while still keeping the number of z_(j)-points inEquation (77) finite. Since

$\begin{matrix}{\int_{- \infty}^{+ \infty}{{\mathbb{d}z^{''}}{\delta_{M}\left( {{z^{''} - {z_{j}\left. \sigma \right)}} = 1} \right.}}} & (79)\end{matrix}$we then have that

$\begin{matrix}\left\langle {z{{{\left( {G_{0}V} \right)^{n}\left. k \right\rangle} = {\sum\limits_{j}^{\;}\left\{ {\left\langle {z{G_{0}}z_{j}} \right\rangle\left\langle {z_{j}\left. {\left( {G_{0}V} \right)^{n - 1}\left. k \right\rangle} \right\}{V\left( z_{j} \right)}} \right.} \right.}}}} \right. & (80)\end{matrix}$

This is clearly tantamount to making a delta function approximation forV(z). Repeating this process for <z_(j)|(G₀V)^(n−1)|k> etc., we canultimately reduce the Born expansion of the wavefunction to a formequivalent to that obtained starting with a interaction that is a sum ofdelta-functions. In this sense the delta-function interaction is quitegeneral, but its application (i.e., in the context of the abovediscussion, fixing u and the number of sample points) sensitivelydepends on the particular problem. From the basic theory of Fouriertransforms, we know that the variation of the integrand in Equation (78)is controlled by its band-width in Fourier transform space, and, inturn, the band-width controls the width of the spacing betweenz_(j)-points through the Nyquist relationship. Clearly one wants theband width to be as small as can be reasonably chosen in order to makethe sample point spacing as large possible.

At the same time the above discussion, also makes clear that one alsowants to minimize the spread of the delta-function approximation aboutthe point z_(j) in Equation (78). The problem of simultaneouslyminimizing the band-width and the spread of the delta-functionapproximation in z-space is addressed through the choice of thedelta-function approximation of the form used in Equation (74). It hasbeen shown elsewhere that, in the sense of the Heisenberg UncertaintyPrinciple, the HDAF approximation (i.e., δ_(M)(z−z_(j)|σ)) is the bestone can do [13A].

The above discussion examines the conditions under which a interactionthat is a sum of delta-functions can adequately represent the trueinteraction. However, it does not justify the procedure of Equation (61)per se since this averaging procedure, which is strictly correct for adelta-function interaction, depends explicitly on the fact that thedelta function interaction is not well tempered. Precisely how thisprocedure should be implemented/modified for a realistic interaction isa subject for further research.

REFERENCES

-   [1A] T. S. Ho and H. Rabitz, J. Chem. Phys. 90, 5614 (1988) and    ibid. 91, 7590 (1989); J. M. Geremia and H. Rabitz, Phys. Rev. A,    64, 022710-1-13 (2001); H. Rabitz, Theor. Chem. Accnts. 109 64    (2003).-   [2A] R. Jost and W. Kohn, Phys. Rev. 87, 977 (1952).-   [3A] H. E. Moses, Phys. Rev. 102. 559 (1956).-   [4A] M. Razavy, J. Acoust. Soc. Am. 58, 956 (1975).-   [5A] R. T. Prosser, J. Math. Phys. 10, 1819 (1969), ibid.17,    1775 (1976) and ibid. 21, 2648 (1980).-   [6A] A. B. Weglein, F. A. Gasparotto, P. M. Carvalho and R. H.    Stolt, Geophys. 62, 1975 (1997).-   [7A] D. J. Kouri, and A. Vijay, Phys. Rev. E 67, 046614-1-12    (2003); D. J. Kouri, A. Vijay and D. K. Homan, J. Phys. Chem. A107,    7230 (2003).-   [8A] R. G. Newton, Scattering Theory of Waves and Particles    (Springer-Verlag, New York, 1982).-   [9A] D. K. Homan and D. J. Kouri to be published.-   [10A] L. S. Rodberg and R. M. Thaler, The Quantum Theory of    Scattering (Academic Press, New York, 1967) pp. 149–153.-   [11A] D. K. Homan and D. J. Kouri in Proc. 3rd Int. Conf. on Math.    and Num. Aspects of Wave Propagation. (SIAM, Philadelphia, 1995) pp.    56–83: see also C. Chandler and A. Gibson, J. Approx. Theory 100.    233 (1999).-   [12A] M. L. Goldberger amd K. M. Watson, Collsion Theory.(Wiley New    York, 1964).-   [13A] D. K. Homan and D. J. Kouri, Phys. Rev. Lett. 85, 5263 (2000);    Phys. Rev. A65, 052106–1 (2002); D. J. Kouri, M. Papadakis, I    Kakadiaris and D. K. Homan, J. Phys. Chem. A107, 7318 (2003).

Inverse Scattering Theory: Renormalization of the Lippmann-SchwingerEquation for Quantum Elastic Scattering with Spherical Symmetry

VII. Introduction

There have been several general approaches to the inverse scatteringproblem in quantum mechanics. The earlier of those was pioneered by Jostand Kohn[1B] and Moses[2B] and it is based on the Born-Neumannperturbation expansion of the Lippmann-Schwinger integral equationdescribing quantum scattering. Additional work on the approach includesthat of Razavey[3B], Prosser[4B], and most importantly, in the contextof the seismic inverse problem, by Weglein and co-workers [5B]. The keymathematical issue in the approach concerns the convergence of theresulting inverse scattering series, and this can be deferred, at leastfor some aspects of the problem, by considering certain subseries[5B].

The other general approach has been that pursued e.g., by Marchenko[6B]and R. G. Newton[7B]. In these approaches, alternative integralequations (of the Volterra-type) are derived leading to extremely robustbehavior under iteration, i.e., absolute convergence independent ofinteraction strength. So far as we can tell, the principle difficultyassociated with these approaches is in the nature of the input datarequired for their implementation. Indeed, it is true in general forquantum scattering that experiment does not readily provide thequantities that are directly involved in the inversion formulae[7B].This is in part a consequence of the fact that in quantum mechanics,probabilities rather than amplitudes are observed, thereby leading toambiguities in phases. The present application is not primarily directedat dealing with this issue, although our results are interesting fromthis aspect. We shall assume that either measurements of angulardistributions are available experimentally since these do provide thesort of phase information that one desires, or that one has access tointegral cross sections for a range of collision energies.

The approach which we shall pursue has its origin in the first class ofmethods [1B–5B]. These methods are most simply formulated in terms ofthe solution, by iteration, of the Lippmann-Schwinger equation for thetransition amplitude. Thus, for structureless particle scattering, in3-D, one has

$\begin{matrix}{T = {V + {{VG}_{0}^{+}T}}} & (81)\end{matrix}$where

$\begin{matrix}{G_{0}^{+} = \frac{1}{E - T - {\mathbb{i}ɛ}}} & (82) \\{H = {T + V}} & (83)\end{matrix}$denotes the non-interacting Greens function and the Hamiltonian, H, isthe sum of T, the kinetic energy, and V, the interaction responsible forthe scattering. We view Equation (81) now as an integral equation for V(rather than for T):

$\begin{matrix}{V = {T - {{VG}_{0}^{+}T}}} & (84)\end{matrix}$

Then a power series solution for V in terms of T has the form

$\begin{matrix}{V = {{T - {{TG}_{0}^{+}T} + {{TG}_{0}^{+}{TG}_{0}^{+}T} - \ldots} = {\sum\limits_{j}^{\;\infty}V_{j}}}} & (85)\end{matrix}$Such an expression is problematic since it requires knowledge of theoff-shell T-matrix elements (which are generally not available sincethey are equivalent to near-field measurements of the wavefunction).However, for the case of a local potential, we must interpret Equation(85) as a sum of local, effective interactions which (provided theseries converges) add up to the true, local interaction. Thus, considerthe first order term:V₁=T  (86)an arbitrary off-shell matrix element of this then is of the form

$\begin{matrix}{\left\langle {\overset{\rightarrow}{k^{\prime}}{V_{1}}\overset{\rightarrow}{k}} \right\rangle = \left\langle {\overset{\rightarrow}{k^{\prime}}{T}\overset{\rightarrow}{k}} \right\rangle} & (87) \\{= {\frac{1}{\left( {2\pi} \right)^{3}}{\int{{\mathbb{d}\;\overset{\rightarrow}{r}}\; e^{{- {\mathbb{i}}}\;{{\overset{\rightarrow}{k}}^{\prime} \cdot \overset{\rightarrow}{r}}}{V_{1}\left( \overset{\rightarrow}{r} \right)}{\mathbb{e}}^{{- {\mathbb{i}}}\;{\overset{\rightarrow}{k} \cdot \overset{\rightarrow}{r}}}}}}} & (88) \\{= {{\overset{\sim}{V}}_{1}\left( {\overset{\rightarrow}{k} - {\overset{\rightarrow}{k}}^{\prime}} \right)}} & (89)\end{matrix}$

Thus, due to the local character assumed for V (and therefore also forV_(j), j=1, 2, . . . ), we can obtain all needed T-matrix elements forEquation (85) once V₁({right arrow over (r)}) is determined [1B–5B].This results from the inverse Fourier transform of Equation (89), wherein particular, we consider backward scattered amplitudes[7B]. Then{right arrow over (k)}′=−{right arrow over (k)} and{tilde over (V)} ₁(2{right arrow over (k)})=<−{right arrow over(k)}|T|{right arrow over (k)}>  (90)V ₁({right arrow over (r)})=2∫d{right arrow over (k)}e ⁻² {right arrowover (k)}·{right arrow over (r)} {tilde over (V)} ₁(2{right arrow over(k)})  (91)Notice that all of the matrix elements of T can be gotten from the{tilde over (V)}₁(2{right arrow over (k)}), simply from the condition<{right arrow over (k)}′|T|{right arrow over (k)}″>={tilde over (V)}₁(2{right arrow over (k)})  (92)where

$\begin{matrix}{\overset{\rightarrow}{k} = {\frac{1}{2}\left( {k^{''} - k^{\prime}} \right)}} & (93)\end{matrix}$

Thus, Equation (85) can also be expressed in the form

$\begin{matrix}{V = {V_{1} - {V_{1}G_{0}^{+}V_{1}} + {V_{1}G_{0}^{+}V_{1}G_{0}^{+}V_{1}} - \ldots}} & (94)\end{matrix}$This is all well and good except that it can only lead to well-definedresults if the perturbation expansion Equation (89) converges.Unfortunately, this is extremely difficult to ascertain in general andit depends on the strength of interaction, V, the existence of boundstates in the spectrum of H and the energy of the collision process,etc. [7B]. In general, the expansion does not converge if theinteraction is too strong (or if it supports bound states).

The goal of the present application is to provide an alternative inversescattering series approach which is guaranteed to converge absolutely,independent of the strength of the interaction. The application isorganized as follows. In the next section, we derive a renormalizedinverse scattering series and discuss its convergence. In section IX, wediscuss the information required to apply the new inversion and insection X we illustrate the approach by applying it to a simple modelscattering system. In section XI, we discuss our results.

VIII. Renormalization of the Lippmann-Schwinger Equation

We begin by remembering that Equations (81)–(94) also apply in anappropriately modified form if one considers the various partial wavecomponents. For simplicity, we shall restrict ourselves to sphericallysymmetric interactions in this application, but the method is general[8B]. The radial Lippmann-Schwinger equation is well known to be [9B]

$\begin{matrix}\begin{matrix}{{\psi_{lk}^{+}(r)}\; = {{j_{l}({kr})} - {\frac{2{mk}}{\hslash}{\int_{0}^{\infty}{{\mathbb{d}r^{\prime}}r^{\prime 2}{h_{l}^{+}\left( {kr}_{>} \right)}{j_{l}\left( {kr}_{<} \right)}{V\left( r^{\prime} \right)}{\psi_{lk}^{+}\left( r^{\prime} \right)}}}}}} \\{= {{j_{l}({kr})} + {\int_{0}^{\infty}{{\mathbb{d}r^{\prime}}r^{\prime 2}{G_{l0k}^{+}\left( {r,r^{\prime}} \right)}{V\left( r^{\prime} \right)}{\psi_{lk}^{+}\left( r^{\prime} \right)}}}}}\end{matrix} & (95)\end{matrix}$where r_(>)(r_(≦)) is the usual greater (lesser) of the pair (r,r′),j_(l) is the l^(th) regular spherical Bessel function,

h_(l)⁺is the l^(th) spherical Hankel function with outgoing wave condition,

ψ_(lk)⁺is the l^(th) partial wave component of the scattering boundarycondition solution to the Schrodinger equation, and

G_(l0k)⁺is defined by the second equality in Equation (95). Specifically,

$\begin{matrix}{{\psi_{\overset{\rightarrow}{k}}^{+}\left( \overset{\rightarrow}{r} \right)} = {{\mathbb{e}}^{{\mathbb{i}}\;{\overset{\rightarrow}{k} \cdot \overset{\rightarrow}{r}}} - {\frac{1}{4\pi}{\int{{\mathbb{d}\;{\overset{\rightarrow}{r}}^{\prime}}\frac{{\mathbb{e}}^{{\mathbb{i}}\;\overset{\rightarrow}{k}{{\overset{\rightarrow}{r} - {\overset{\rightarrow}{r}}^{\prime}}}}}{{\overset{\rightarrow}{r} - {\overset{\rightarrow}{r}}^{\prime}}}{V\left( {\overset{\rightarrow}{r}}^{\prime} \right)}{\psi_{\overset{\rightarrow}{k}}^{+}\left( {\overset{\rightarrow}{r}}^{\prime} \right)}}}}}} & (96) \\{{\mathbb{e}}^{{\mathbb{i}}\;{\overset{\rightarrow}{k} \cdot \overset{\rightarrow}{r}}} = {\sum\limits_{l}^{\;}{\sum\limits_{m}^{\;}\;{i^{l}{Y_{lm}\left( \hat{r} \right)}{Y_{lm}^{*}\left( \hat{k} \right)}{j_{l}({kr})}}}}} & (97) \\{{{- \frac{1}{4\pi}}\frac{{\mathbb{e}}^{{\mathbb{i}}\; k{{\overset{\rightarrow}{r} - {\overset{\rightarrow}{r}}^{\prime}}}}}{{\overset{\rightarrow}{r} - {\overset{\rightarrow}{r}}^{\prime}}}} = {{- \frac{2{mk}}{\hslash^{2}}}{\sum\limits_{l}^{\;}{\sum\limits_{m}^{\;}{{Y_{l}^{m}\left( \hat{r} \right)}{Y_{l}^{m}\left( {\hat{r}}^{\prime} \right)}^{*}{h_{l}\left( {kr}_{>} \right)}{j_{l}\left( {kr}_{<} \right)}}}}}} & (98) \\{{\psi_{\overset{\rightarrow}{k}}^{+}\left( \overset{\rightarrow}{r} \right)} = {\sum\limits_{l}^{\;}{\sum\limits_{m}^{\;}\;{i^{l}{Y_{lm}\left( \hat{r} \right)}{Y_{lm}^{*}\left( \hat{k} \right)}{\psi_{lk}^{+}(r)}}}}} & (99)\end{matrix}$The asymptotic form of

ψ_(lk)⁺(r)(for any r>r_(max) such that V(r)=0) is

$\begin{matrix}\left. {\psi_{lk}^{+}(r)}\rightarrow{{j_{l}({kr})} + {T_{l}^{(1)}{h_{l}^{+}({kr})}}} \right. & (100)\end{matrix}$where

$\begin{matrix}{T_{l}^{(1)} \equiv {{- \frac{2{mk}}{\hslash^{2}}}{\int_{0}^{\infty}{{\mathbb{d}{rr}^{2}}{j_{l}({kr})}{V(r)}{\psi_{lk}^{+}(r)}}}}} & (101)\end{matrix}$With this definition, the (unitary) S-matrix, S_(l), satisfiesS _(l)=1+2iT _(l) ⁽¹⁾  (102)implying thatT _(l) ⁽¹⁾ =e ^(iη) ^(l) sin η_(l)  (103)Here η_(l) is the usual phase shift, andS_(l)=e^(iη) ^(l)   (104)It will also prove necessary to define an additional quantity:

$\begin{matrix}{T_{l}^{(2)} \equiv {{- \frac{2{mk}}{\hslash^{2}}}{\int_{0}^{\infty}\ {{\mathbb{d}{rr}^{2}}{\eta_{l}({kr})}{V(r)}{\psi_{lk}^{+}(r)}}}}} & (105)\end{matrix}$

We note that in general, T_(l) ⁽²⁾ is not a directly measured quantitynor is it immediately obtainable from measured quantities. Finally, thedifferential scattering amplitude, f(θ), is

$\begin{matrix}{{f(\theta)} = {\frac{1}{k}{\sum\limits_{l}^{\;}\;{\left( {{2l} + 1} \right){P_{l}\left( {\cos\;\theta} \right)}T_{l}}}}} & (106)\end{matrix}$where θ is the angle between the incident relative momentum vector,{right arrow over (k)}, and the direction of observation, {circumflexover (r)}. For mathematical simplicity, we shall also assume that theinteraction has “compact support”, i.e., it is zero outside the ranger_(max):V(r)=0, r>r _(max)  (107)In general, however, our results will hold for interactions that are nottoo singular at r=0 and that tend to zero faster than 1/r, as r→∞.Following Sams and Kouri[10B] and Kouri and Vijay[11B], we rewriteEquation (95) as

$\begin{matrix}\begin{matrix}{{\psi_{lk}^{+}(r)} = {{{j_{l}({kr})}\left\lbrack {1 - {\frac{2{mk}}{\hslash^{2}}{\int_{- \infty}^{+ \infty}\ {{\mathbb{d}r^{\prime}}r^{\prime 2}{h_{l}^{+}\left( {kr}^{\prime} \right)}{V\left( r^{\prime} \right)}{\psi_{lk}^{+}\left( r^{\prime} \right)}}}}} \right\rbrack} -}} \\{\frac{2{mk}}{\hslash^{2}}{\int_{0}^{r}\ {{\mathbb{d}r^{\prime}}{r^{\prime 2}\left\lbrack {{{n_{l}({kr})}{j_{l}\left( {kr}^{\prime} \right)}} - {{j_{l}({kr})}{n_{l}\left( {kr}^{\prime} \right)}}} \right\rbrack}{V\left( r^{\prime} \right)}{\psi_{lk}^{+}\left( r^{\prime} \right)}}}}\end{matrix} & (108)\end{matrix}$Buth _(l) ⁺+(kr′)=η_(l)(kr)+ij _(l)(kr)  (109)so we write Equation (108) as

$\begin{matrix}\begin{matrix}{{\psi_{lk}^{+}(r)} = {{{j_{l}({kr})}\left\lbrack {1 + T_{l}^{(2)} + {{\mathbb{i}}\; T_{l}^{(1)}}} \right\rbrack} -}} \\{\frac{2{mk}}{\hslash^{2}}{\int_{0}^{\infty}\ {{\mathbb{d}r^{\prime}}{r^{\prime 2}\left\lbrack {{{\eta_{l}({kr})}{j_{l}\left( {kr}^{\prime} \right)}} - {{j_{l}({kr})}{\eta_{l}\left( {kr}^{\prime} \right)}}} \right\rbrack}{V\left( r^{\prime} \right)}{\psi_{lk}^{+}\left( r^{\prime} \right)}}}}\end{matrix} & (110)\end{matrix}$

We recognize that the factor, [1+T_(l) ⁽²⁾+iT_(l(1))], while unknown, issimply a constant normalization, so that

$\begin{matrix}{{\psi_{lk}^{+}(r)} = {{u_{lk}(r)}\left\lbrack {1 + T_{l}^{(2)} + {{\mathbb{i}}\; T_{l}^{(1)}}} \right\rbrack}} & (111)\end{matrix}$where

$\begin{matrix}{{u_{lk}(r)} = {{j_{l}({kr})} + {\int_{0}^{r}\ {{\mathbb{d}r^{\prime}}r^{\prime 2}{{\overset{\sim}{G}}_{l0k}\left( {r,r^{\prime}} \right)}{V\left( r^{\prime} \right)}{u_{kl}\left( r^{\prime} \right)}}}}} & (112) \\{{{\overset{\sim}{G}}_{l0k}\left( {r,r^{\prime}} \right)} = {- {\frac{2{mk}}{\hslash^{2}}\left\lbrack {{{\eta_{l}({kr})}{j_{l}\left( {kr}^{\prime} \right)}} - {{j_{l}({kr})}{\eta_{l}\left( {kr}^{\prime} \right)}}} \right\rbrack}}} & (113)\end{matrix}$

Equation (112) for u_(lk)(r) has the tremendous virtue, compared to theLippmann-Schwinger equation for

ψ_(lk)⁺(r),of being a Volterra integral equation and under iteration it convergesabsolutely and uniformly for all appropriately measurable interactions.This is because the kernel, {tilde over (G)}_(l0k) (r, r′)V(r′), istriangular, implying that the Fredholm determinant is identicallyone[7B]. Consequently, it has no zeros and the Fredholm solutionencounters no singular points. This is the best possible mathematicalsituation one can ever have!

However, we still must address the problem of how to make use ofEquation (111), since T_(l) ⁽²⁾ is not readily available. Before dealingwith this, we note that in analogy with our earlier work on acousticscattering [11 B], we can introduce a partial wave transition operator,T_(l):

$\begin{matrix}{{V\;\psi_{lk}^{+}} = {T_{l}j_{l}}} & (114)\end{matrix}$

$\begin{matrix}{T_{l} = {V + {V\; G_{l0k}^{+}T_{l}}}} & (115)\end{matrix}$and the Volterra-based auxiliary operators:

$\begin{matrix}{{\overset{\sim}{G}}_{l0k} = {G_{l0k}^{+} + {\frac{2{mk}}{\hslash^{2}}\left. j_{l} \right\rangle\left\langle h_{l}^{+} \right.}}} & (116)\end{matrix}$or in the coordinate representation,

$\begin{matrix}{{{\overset{\sim}{G}}_{l0k}\left( {r,r^{\prime}} \right)} = {{G_{l0k}^{+}\left( {r_{<},r_{>}} \right)} + {\frac{2{mk}}{\hslash^{2}}{j_{l}({kr})}{h_{l}^{+}\left( {kr}^{\prime} \right)}}}} & (117)\end{matrix}$Then we define {tilde over (T)}_(l) such thatVu _(kl) ={tilde over (T)} _(l) j _(l)  (118){tilde over (T)} _(l) =V+V{tilde over (G)} _(l0k) {tilde over (T)}_(l)  (119)We see that

$\begin{matrix}{T_{l} = {{\overset{\sim}{T}}_{l}\left( {1 - {\frac{2{mk}}{\hslash^{2}}\left. j_{l} \right\rangle\left\langle h_{l}^{+} \right.T_{l}}} \right)}} & (120) \\{\mspace{20mu}{= {{\overset{\sim}{T}}_{l}\left( {1 - {\frac{2{mk}}{\hslash^{2}}\left. j_{l} \right\rangle\left\langle \eta_{l} \right.T_{l}} - {\frac{2{mk}}{\hslash^{2}}\left. j_{l} \right\rangle\left\langle j_{l} \right.T_{l}}} \right)}}} & (121)\end{matrix}$It follows that

$\begin{matrix}\begin{matrix}{T_{l}^{(1)} = {{\overset{\sim}{T}}_{l}^{(1)}\left\lbrack {1 + T_{l}^{(2)} + {{\mathbb{i}}\; T_{l}^{(1)}}} \right\rbrack}} \\{T_{l}^{(2)} = {{\overset{\sim}{T}}_{l}^{(2)}\left\lbrack {1 + T_{l}^{(2)} + {{\mathbb{i}}\; T_{l}^{(1)}}} \right\rbrack}}\end{matrix} & (122)\end{matrix}$We then see that

$\begin{matrix}{T_{l}^{(1)} = \frac{{\overset{\sim}{T}}_{l}^{(1)}}{1 - {\overset{\sim}{T}}_{l}^{(2)} - {{\mathbb{i}}{\overset{\sim}{T}}_{l}^{(1)}}}} & (123)\end{matrix}$This relation enables us to express the perturbation expansion of V interms of {tilde over (T)}_(l) ⁽¹⁾ and {tilde over (T)}_(l) ⁽²⁾thenultimately in terms of {tilde over (T)}_(l) ⁽¹⁾. We stress, however,that from Equation (119),

$\begin{matrix}{V = {{\overset{\sim}{T}}_{l} - {V{\overset{\sim}{G}}_{l0k}{\overset{\sim}{T}}_{l}}}} & (124) \\{\mspace{20mu}{= {{\overset{\sim}{T}}_{l} - {{\overset{\sim}{T}}_{l}{\overset{\sim}{G}}_{l0k}{\overset{\sim}{T}}_{l}} + {{\overset{\sim}{T}}_{l}{\overset{\sim}{G}}_{l0k}{\overset{\sim}{T}}_{l}{\overset{\sim}{G}}_{l0k}{\overset{\sim}{T}}_{l}} + \cdots}}} & (125)\end{matrix}$which expression converges absolutely and uniformly independent of thestrength of the interaction. We shall again restrict ourselves to local(and for this application, spherically symmetric) interactions, V(r).

We now consider how to determine the {tilde over (V)}_(j), defined by

$\begin{matrix}{V = {\sum\limits_{j = 1}^{\infty}V_{j}}} & (126)\end{matrix}${tilde over (V)}₁={tilde over (T)}₁  (127){tilde over (V)} ₂ =−{tilde over (V)} ₁ {tilde over (G)} _(l0k) {tildeover (V)} ₁  (128){tilde over (V)} ₃={tilde over (V)}₁ {tilde over (G)} _(l0k) {tilde over(V)} ₁ {tilde over (G)} _(l0k) {tilde over (V)} ₁ =− {tilde over (V)} ₂{tilde over (G)} _(l0k) {tilde over (V)} ₁  (129){tilde over (V)} _(j) ={tilde over (V)} _(j−1) {tilde over (G)} _(l0k){tilde over (V)} ₁  (130)

-   -    

$\begin{matrix}{{\overset{\sim}{V}}_{1l}^{(1)} \equiv {{- \frac{2{mk}}{\hslash^{2}}}{\int_{0}^{\infty}{{\mathbb{d}r}\; r^{2}\;{j_{l}^{2}({kr})}\;{{\overset{\sim}{V}}_{1}(r)}}}}} & (131) \\{{\overset{\sim}{V}}_{1l}^{(2)} \equiv {{- \frac{2{mk}}{\hslash^{2}}}{\int_{0}^{\infty}{{\mathbb{d}r}\; r^{2}\;{\eta_{l}({kr})}\;{{\overset{\sim}{V}}_{1}(r)}\;{j_{l}({kr})}}}}} & (132)\end{matrix}$

We have written the upper limit as ∞ rather than r_(max) in anticipationof the fact that, provided {tilde over (V)}₁(r) tends to zero fasterthan 1/r², the more general result holds. It is not difficult then toshow that

$\begin{matrix}{T_{l}^{(1)} = \frac{{\overset{\sim}{V}}_{1l}^{(1)}}{1 - {\overset{\sim}{V}}_{1l}^{(2)} - {{\mathbb{i}}{\overset{\sim}{V}}_{1l}^{(1)}}}} & (133) \\{T_{l}^{(2)} = \frac{{\overset{\sim}{V}}_{2l}^{(1)}}{1 - {\overset{\sim}{V}}_{1l}^{(2)} - {{\mathbb{i}}{\overset{\sim}{V}}_{1l}^{(1)}}}} & (134)\end{matrix}$

These results have some extremely interesting features. First, note thatboth {tilde over (V)}_(1l) ⁽¹⁾ and {tilde over (V)}_(1l) ⁽²⁾ are purelyreal. Consequently, Equation (133) guarantees satisfaction of theoptical theorem, since

$\begin{matrix}{{{Im}\mspace{11mu} T_{l}^{(1)}} = {T_{l}^{(1)}}^{2}} & (135)\end{matrix}$(This also implies that one cannot eliminate V_(1l) ⁽²⁾ in favor ofV_(1l) ⁽¹⁾ by using the real and imaginary parts of Equation (133)).Second, Equation (133) is sufficient to enable an inversion, providedthat T_(l)(1) is known. Thus, Equation (133) is viewed as an equationthat is satisfied by the first order radial function, {tilde over(V)}₁(r):

$\begin{matrix}{{T_{l}^{(1)}(k)} = \frac{{- \frac{2{mk}}{\hslash^{2}}}{\int_{0}^{\infty}{{\mathbb{d}r}\; r^{2}{j_{l}^{2}({kr})}\;{{\overset{\sim}{V}}_{1}(r)}}}}{\begin{matrix}{{1 + {\frac{2{mk}}{\hslash^{2}}{\int_{0}^{\infty}{{\mathbb{d}{rr}^{2}}{\eta_{l}({kr})}{{\overset{\sim}{V}}_{1}(r)}{j_{l}({kr})}}}} +}} \\{\frac{2{mk}}{\hslash^{2}}{\int_{0}^{\infty}{{\mathbb{d}{rr}^{2}}\;{j_{l}^{2}({kr})}\;{{\overset{\sim}{V}}_{1}(r)}}}}\end{matrix}}} & (136)\end{matrix}$

Now, of course, the inversion is not so simple since one no longer hasCartesian Fourier transforms to invert. One procedure is to express{tilde over (V)}₁(r) in some basis set (e.g., Bessel functions); analternative is to use the distributed approximating functionals. In anycase, it may be necessary to obtain {tilde over (V)}₁(r) on a numericalgrid. We conclude that the fundamental results are Equations(133)–(134), along with:{tilde over (V)} _(j+1) =−{tilde over (V)} _(j) {tilde over (G)} _(l0k){tilde over (V)} ₁, j≧1  (137)We now turn to discuss the implementation of this approach in terms ofmeasurable quantities.IX. Experimental Data Requirements for Implementation of theVolterra-Based Inverse Series

In quantum mechanical elastic scattering, the optimum measurements arethe differential angular distributions, which are determined by |f(θ)|².Due to azimuthal symmetry, we, in fact, consider 2π|f(θ)|² sin θ:

$\begin{matrix}{\frac{\partial{\sigma(\theta)}}{\partial\theta} = {\sin\mspace{11mu}\theta\;\frac{2\;\pi}{k^{2}}{{\sum\limits_{l}{\left( {{2l} + 1} \right)\mspace{11mu}{P_{l}\left( {\cos\mspace{11mu}\theta} \right)}T_{l}^{(1)}}}}^{2}}} & (138)\end{matrix}$Clearly, if sufficient number of scattering angles are measured, one can(in principle) determine the T_(l) ⁽¹⁾. An alternative that avoidshaving to determine the individual T_(l) ⁽¹⁾ is to use Equation (136)directly, along with an appropriate representation of {tilde over(V)}₁(r) to derive a system of inhomogeneous (non-linear) algebraicequations which can be solved. For example, if we expand {tilde over(V)}₁(r) in a basis, {φ_(p)(r)},

$\begin{matrix}{{{\overset{\sim}{V}}_{1}(r)} = {\sum\limits_{p}{{\overset{\sim}{V}}_{1p}{\phi_{p}(r)}}}} & (139)\end{matrix}$it can then be seen that

$\begin{matrix}{\frac{\partial{\sigma(\theta)}}{\partial\theta} = {\sin\mspace{11mu}\theta\;\frac{2\;\pi}{k^{2}}{\frac{\sum\limits_{l}{\left( {{2l} + 1} \right)\mspace{11mu}{P_{l}\left( {\cos\mspace{11mu}\theta} \right)}\;{\sum\limits_{p}{{\overset{\sim}{V}}_{1p}J_{1p}}}}}{1 - {\sum\limits_{p^{\prime}}{{\overset{\sim}{V}}_{1p^{\prime}}H_{1p^{\prime}}}}}}^{2}}} & (140) \\{where} & \; \\{J_{lp} = {{- \frac{2{mk}}{\hslash^{2}}}{\int_{0}^{\infty}{{\mathbb{d}r}\; r^{2}\;{j_{l}^{2}({kr})}\;{\phi_{p}(r)}}}}} & (141) \\{and} & \; \\{H_{lp} = {{- \frac{2{mk}}{\hslash^{2}}}{\int_{0}^{\infty}{{\mathbb{d}r}\; r^{2}\;{h_{l}^{+}({kr})}\;{j_{l}({kr})}\;{\phi_{p}(r)}}}}} & (142)\end{matrix}$In general, Equation (140) would be solved by a least square method,using more θ-values than the number of terms in the expansion over p,Equation (139). The redundancy is useful for averaging out noise.Another alternative is to use angular measurements at a small number ofθ's, but for a range of collision energies (E=

²k²/2m) to obtain an over-determined set of simultaneous nonlinearalgebraic equations to solve.

Another interesting approach can be based on integral cross sectionmeasurements. Thus, the integral cross section at energy E is well-knownto be

$\begin{matrix}{{\sigma(E)} = {\frac{4\;\pi}{k^{2}}{\sum\limits_{l}{\left( {{2l} + 1} \right)\;{T_{l}}^{2}}}}} & (143)\end{matrix}$leading to the expression

$\begin{matrix}{{\sigma(E)} = {\frac{4\;\pi}{k^{2}}{\sum\limits_{l}{\left( {{2l} + 1} \right)\;{\frac{\sum\limits_{p}{{\overset{\sim}{V}}_{1p}J_{lp}}}{1 - {\sum\limits_{p^{\prime}}{{\overset{\sim}{V}}_{1p^{\prime}}H_{{lp}^{\prime}}}}}}^{2}}}}} & (144)\end{matrix}$One must evaluate σ(E) at enough energies to generate the requisitealgebraic equations for the {tilde over (V)}_(1p). In expression (140)and (144), it is clear that lower energy measurements will benumerically less complicated because the partial wave expansion willconverge with fewer angular momentum states. However, we also expectthat (at least for potentials with a repulsive core) the short rangepart of the potential will be less accurate than the longer range, ifone uses low energy data.

It is important to note that in the case of the integral cross sectionapproach, one makes no use of phase-dependent effects; indeed, only the|T_(l)|² enter the expression. Neither the basis set nor DAF approachesnecessarily require knowledge of the phase of the T_(l), but onespeculates that an inversion based on angular measurements will be morerobust (in terms of accuracy) than one based on integral cross sectionmeasurements. This remains to be tested. If either the integral ordifferential cross section approach proves to be feasible, this willrepresent an extremely attractive feature compared to approaches thatrequire determination of the individual partial wave phase-shifts.

Finally, we point out that once {tilde over (V)}_(l)(r) is known, allhigher order {tilde over (V)}_(j)(r) can be computed in the coordinaterepresentation, using Equation (137). We notice that the same radialfunctions, {tilde over (V)}_(j)(r), result no matter which partial waveis considered. This is a consequence of the assumed spherical symmetryof the original potential. This provides an internal consistencycondition that must be satisfied.

Of course, the above ideas, while formally correct, still must be testedon actual experimental data. An important issue is that of the effectsof noise and inaccuracies in the data. In this regard, DAF-based methodsmay offer advantages. In a DAF approach, an approximation to theidentity, of the form

$\begin{matrix}{{\delta_{M}\left( {r - r^{\prime}} \middle| \sigma \right)} = {\sum\limits_{n = 0}^{M}{{\phi_{n}(0)}\;{\phi_{n}^{*}\left( \frac{r - r^{\prime}}{\sigma} \right)}}}} & (145)\end{matrix}$is employed. Generally, it has the properties that

$\begin{matrix}{\left. {\lim\limits_{M\rightarrow\infty}{{\delta_{M}\left( {r - r^{\prime}} \right.}\sigma}} \right) = {\delta\left( {r - r^{\prime}} \right)}} & (146)\end{matrix}$and

$\begin{matrix}{{\lim\limits_{\sigma->0}{\delta_{M}\left( {r - r^{\prime}} \middle| \sigma \right)}} = {\delta\left( {r - r^{\prime}} \right)}} & (147)\end{matrix}$here o is a length-scale parameter. For finite M or nonzero σ,δ_(M)(r−r′|σ) is not a projector onto an orthogonal subspace of Hilbertspace (though it is, usually, such a projector onto the Schwartzsubspace). The DAF approximation (which can be made arbitrarilyaccurate) to the potential is then

$\begin{matrix}{{{\overset{\sim}{V}}_{1}(r)} = {\int_{0}^{\infty}{{\mathbb{d}r^{\prime}}r^{\prime\; 2}\mspace{11mu}{\delta_{M}\left( {r - r^{\prime}} \middle| \sigma \right)}\;{{\overset{\sim}{V}}_{1}\left( r^{\prime} \right)}}}} & (148) \\{\mspace{56mu}{= {\sum\limits_{n = 0}^{M}{{\phi_{n}(0)}\;{\int_{0}^{\infty}{{\mathbb{d}r^{\prime}}\; r^{\prime\; 2}\mspace{11mu}{\phi_{n}^{*}\left( \frac{r - r^{\prime}}{\sigma} \right)}\;{{\overset{\sim}{V}}_{1}\left( r^{\prime} \right)}}}}}}} & (149)\end{matrix}$

If we evaluate the integral by quadrature, then

$\begin{matrix}{{{\overset{\sim}{V}}_{1}(r)} \equiv {\sum\limits_{p}{W_{p}r_{p}^{2}{\delta_{M}\left( {r - r_{p}} \middle| \sigma \right)}\;{\overset{\sim}{V}}_{1p}}}} & (150)\end{matrix}$and the {tilde over (V)}_(1p) in this approach are seen to be thediscrete, point-wise samples of the first order potential. Clearly, onewill obtain expressions for dσ/dθ and σ(E) that are entirely analogousto those resulting from the basis set expansion approach. Both methodsrequire the solution of nonlinear algebraic equations and the sameexperimental data is employed.X. A Simple Example

By far the simplest scattering problem to solve from the point of viewof the Lippmann-Schwinger equation is for a local, Dirac delta functionpotential. In our first study of inverse acoustic scattering, we foundthat the Volterra-based series converged to the exact result in a singleterm. A simple 3-D analogue is the spherically symmetric potentialV(r)=λδ(r−r ₀)  (151)It can then be shown that the exact transition amplitude is

$\begin{matrix}{T_{l}^{(1)} = \frac{{- \frac{2{mk}}{\hslash^{2}}}\lambda\; r_{0}^{2}{j_{l}^{2}\left( {kr}_{0} \right)}}{1 + {\frac{2{mk}}{\hslash^{2}}\lambda\; r_{0}^{2}{\eta_{l}\left( {kr}_{0} \right)}{j_{l}\left( {kr}_{0} \right)}} + {\frac{2{mk}}{\hslash^{2}}\lambda\; r_{0}^{2}{j_{l}^{2}\left( {kr}_{0} \right)}}}} & (152)\end{matrix}$

Given such detailed input, we can use Equation (136) directly:

$\begin{matrix}{\frac{\lambda\; r_{0}^{2}{j_{l}^{2}\left( {kr}_{0} \right)}}{1 + {\frac{2{mk}}{\hslash^{2}}\lambda\; r_{0}^{2}{\eta_{l}\left( {kr}_{0} \right)}{j_{l}\left( {kr}_{0} \right)}} + {\frac{2{mk}}{\hslash^{2}}\lambda\; r_{0}^{2}{j_{l}^{2}\left( {kr}_{0} \right)}}} = \frac{\int_{0}^{\infty}{{\mathbb{d}{rr}^{2}}{j_{l}^{2}({kr})}{{\overset{\sim}{V}}_{1}(r)}}}{1 + {\frac{2{mk}}{\hslash^{2}}{\int_{0}^{\infty}{{\mathbb{d}{rr}^{2}}{\eta_{l}({kr})}{{\overset{\sim}{V}}_{1}(r)}{j_{l}({kr})}}}} + {\frac{2{mk}}{\hslash^{2}}{\int_{0}^{\infty}{{\mathbb{d}{rr}^{2}}{j_{l}^{2}({kr})}{{\overset{\sim}{V}}_{1}(r)}}}}}} & (153)\end{matrix}$Obviously, the solution is (independent of the partial wave considered){tilde over (V)} ₁(r)=λδ(r−r ₀)  (154)

Next, we must evaluate the higher order terms in the expansion of V(r)in terms of the {tilde over (V)}_(j)(r). By Equation (137),

$\begin{matrix}{{{\overset{\sim}{V}}_{2}(r)} = {{- {\overset{\sim}{V}}_{1}}{\overset{\sim}{G}}_{l\; 0k}{\overset{\sim}{V}}_{1}}} & (155) \\{= {{- {{\lambda\delta}\left( {r - r_{0}} \right)}}{\int_{0}^{\infty}{{\mathbb{d}r^{\prime}}{{\overset{\sim}{G}}_{l\; 0k}\left( {r,r^{\prime}} \right)}{{\lambda\delta}\left( {r^{\prime} - r_{0}} \right)}}}}} & (156) \\{= {{- {{\lambda\delta}\left( {r - r_{0}} \right)}}{{\overset{\sim}{G}}_{l\; 0k}\left( {r_{0},r_{0}} \right)}}} & (157)\end{matrix}$Clearly, due to the behavior of {tilde over (G)}_(l0k) (r₀, r₀), {tildeover (V)}₂ (r) is identically zero, no matter what that value of l.Further, by Equation (137), all higher {tilde over (V)}_(j)'s are alsozero. We conclude that

$\begin{matrix}{{V(r)} = {{\sum\limits_{j}{{\overset{\sim}{V}}_{j}(r)}} \equiv {{\lambda\delta}\left( {r - r_{0}} \right)}}} & (158)\end{matrix}$Thus, the Volterra-based inverse series again converges to the exactresult in a single term.

Of course, in general, one does not know the individual T_(l)s. In thismodel problem, the differential scattering amplitude is

$\begin{matrix}{{f(\theta)} = {{- \frac{2{mkr}_{0}^{2}}{\hslash^{2}}}\frac{\sum\limits_{l}{\left( {{2l} + 1} \right){P_{l}\left( {\cos\;\theta} \right)}{j_{l}^{2}\left( {kr}_{0} \right)}}}{1 + {\frac{2{mk}}{\hslash^{2}}\lambda\; r_{0}^{2}{h_{l}^{+}\left( {kr}_{0} \right)}{j_{l}\left( {kr}_{0} \right)}}}}} & (159)\end{matrix}$and the cross section is the square of its modulus. The convergence ofthis partial wave series results from the property of the Besselfunctions, j_(l)(kr₀), that j_(l)(kr₀)→0 for l>kr₀. Again, one can inprinciple obtain sufficient equations and obtain the exact result. Thebasic conclusion is the same, namely that the Volterra inversionconverges to the exact result in a single term.XI. Discussion of Results

In this application we have presented a new approach to the inversescattering problem in quantum mechanics. Although attention was focusedon purely elastic scattering by spherically symmetric potential, themethod is quite general. Indeed, it not only can be applied to quantumscattering, but to many other types of processes. Any process that canbe described by a Lippmann-Schwinger type, causal (or anticausal)integral equation should be amenable to the approach. The method isbased on a renormalization transformation of the Lippmann-SchwingerFredholm equation to obtain a Volterra integral equation. In quantumscattering, such equations are well known but principally used toanalyze the analytic structure of the S-matrix. An exception is theearlier work of Sams and Kouri[10B], who utilized the renormalizationpoint of view to develop a noniterative numerical method for directlysolving for the coordinate representation of the T-matrix. The principalbenefit of the renormalization to a Volterra equation for inversescattering is the fact that their noniterative solutions convergeabsolutely and (under relatively mild conditions) uniformallyindependent of the strength of the interaction. This feature allows usto utilize the Volterra equations in a manner similar to that pioneeredby Jost and Kohn[1B], Moses[2B] and most recently by Weglein[5B], butwith the guarantee that the inverse series always converges.

In the case of quantum scattering in 3-D, the results are complicated bythe facts that (a) the renormalization factor is no longer a directlymeasurable quantity as it is for acoustic scattering in 1-D, (b) thedifferent partial waves do not separate in a simple fashion, (c) theequations which one must solve to determine the potential are nonlinear,due to the intrinsic nature of quantum mechanics. However, there are nodifficulties in principle with the present method. Furthermore, thepresent inverse series does not require the determination of phases. Itcan, at least in principle, be applied either to differential orintegral cross section measurements. If it indeed is the case thatsufficiently accurate results can be obtained without requiringdetermination of phase sensitive quantities, this will provide a majoradvantage over other inversion equations for quantum scattering.

For the case of scattering by a spherically symmetric Dirac deltafunction potential, the convergence to the exact result is obtained witha single term. By contrast, the Born-Neumann inverse series based on theLippmann-Schwinger equation yields the result

$\begin{matrix}{{\int_{0}^{\infty}{{\mathbb{d}{rr}^{2}}{V_{1}(r)}{j_{l}^{2}({kr})}}} = T_{l}^{(1)}} & (160)\end{matrix}$for the first order, effective local interaction. It is immediate thatany real V₁ obtained from the above will introduce unphysical behaviorsince the left hand side of the equation is real; i.e., Equation (160)manifestly violates the optical theorem for real V_(l)(r). Comparingthis to Equation (132) and using (152) leads to

$\begin{matrix}{{{- \frac{2{mk}}{\hslash^{2}}}{\int_{0}^{\infty}{{\mathbb{d}{rr}^{2}}{V_{1}(r)}{j_{l}^{2}({kr})}}}} = \frac{{- \frac{2{mk}}{\hslash^{2}}}\lambda_{0}^{2}{j_{l}^{2}\left( {kr}_{0} \right)}}{1 + {\frac{2{mk}}{\hslash^{2}}r_{0}^{2}\lambda\;{h_{l}^{+}\left( {kr}_{0} \right)}{j_{l}\left( {kr}_{0} \right)}}}} & (161)\end{matrix}$a solution of this equation is seen to be

$\begin{matrix}{{V_{1}(r)} = \frac{{\lambda\delta}\left( {r - r_{0}} \right)}{1 + {\frac{2{mk}}{\hslash^{2}}\lambda\; r_{0}^{2}{h_{l}^{+}\left( {kr}_{0} \right)}{j_{l}\left( {kr}_{0} \right)}}}} & (162) \\{{The}\mspace{14mu}{second}\mspace{14mu}{order}\mspace{14mu}{correction}\mspace{14mu}{is}} & \; \\{{V_{2}(r)} = \frac{{- \lambda^{2}}{\delta\left( {r - r_{0}} \right)}{G_{l\; 0k}^{+}\left( {r_{0},r_{0}} \right)}}{\left\lbrack {1 + {\frac{2{mk}}{\hslash^{2}}\lambda\; r_{0}^{2}{h_{l}^{+}\left( {kr}_{0} \right)}{j_{l}\left( {kr}_{0} \right)}}} \right\rbrack^{2}}} & (163) \\{\neq 0} & (164)\end{matrix}$Thus, in this case, the first order term of the series does not yieldthe exact answer in general, and it does not consist of a single nonzeroterm. In fact, one can then see that one must sum the infinite seriesanalytically in order to obtain the correct result for values of λ thatare outside the convergence limit of the series. Equation (160)corresponds to the first term in the Taylor expansion of the denominatoron the right hand side of Equation (161), which is analogous to thesituation we encountered in our previous work on 1D inverse acousticscattering. Such an expansion converges only for sufficientlysmall-values (as well as also depending on the value of r₀). Of course,it does permit one to sum the infinite series analytically to obtain theresult that holds outside the convergence limits of the series itself[11B]. As is also usual for the Born-Neumann expansion in quantumscattering, the approximation does eventually converge for high enoughenergy, E (large enough k).

We are currently exploring the inversion of quantum 3D elasticscattering by a nonspherical target, as well as various other wavephenomena. Of particular interest are the cases of acoustic andelectromagnetic scattering in full 3D. In addition, we shall carry outtest calculations to verify that one can use non-phase sensitive,integral cross sections to carry out an inversion. These results will bereported as they are obtained.

REFERENCES

-   [1B] R. Jost and W. Kohn, Phys. Rev. 87, 977 (1952).-   [2B] H. E. Moses, Phys. Rev. 102, 559 (1956).-   [3B] M. Razavy, J. Acoust. Soc. Am. 58, 956 (1975).-   [4B] R. T. Prosser, J. Math. Phys. 10, 1819 (1969); ibid., 17, 1775    (1976); ibid., 21, 2648 (1980).-   [5B] A. B. Weglein, K. H. Matson, D. J. Foster, P. M. Carvalho, D.    Corrigan, and S. A. Shaw, Imaging and inversion at depth without a    velocity model: theory, concepts and initial evaluation, Soc.    Exploration Geophysics 2000, Expanded Abstracts, Calgary, C A; A. B.    Weglein and R. H. Stolt, Migration-inversion revisited, in The    Leading Edge (1999) 950. See also the subseries approach to removing    multiples from seismic data in A. B. Weglein, F. A.    Gasparotto, P. M. Carvalho, and R. H. Stolt, Geophysics 62, 1775    (1997).-   [6B] marchenko-   [7B] R. G. Newton, Scattering Theory of Waves and Particles    (Springer-Verlag, New York, (1982).-   [8B] D. J. Kouri (To be published).-   [9B] radial-   [10B] W. N. Sams and D. J. Kouri, J. Chem. Phys. 51, 4809 and 4815    (1969).-   [11B] D. J. kouri and Amrendra Vijay, Phys. Rev. E. (in press).

Inverse Scattering Theory: Renormalization of the Lippmann-SchwingerEquation for Acoustic Scattering in One Dimension

XII. Introduction

The inverse scattering problem has enormous importance both forpractical and theoretical applications. The former include hydrocarbonexploration and production, medial imaging of many varieties,nondestructive testing, target identification and location, etc. Thelatter include relating interactions governing atomic and molecularsystems to experimental measurements, determination of the structure ofsurfaces and condensed matter systems, imaging of nanostructures, etc.In much of the literature, the focus has been on determining theconditions under which the data inversion will yield a unique result andprecisely what information is required to make an inversion possible. Interms of algorithms employed for various types of imaging, an importantpractical tool is the first Born approximation, which assumes that allscattering is direct, involving a single interaction of the probe withthe target. Of course, this is known to be incorrect. Indeed, mostimaging procedures or algorithms typically make use of some assumedmodel for the propagation of the probe signal or disturbance in thescattering medium.

Generally, inversion is practical only in their circumstance that thereis a sufficiently small difference between the propagation of the probesignal within the target and its “reference propagation” (low contrastbetween the target and the reference medium). Over the last decade,Weglein and co-workers [1C] have pioneered inverse acoustic scatteringmethods that do not require an assumed propagation velocity model withinthe medium. Their approach is based on the early work of Jost and Kohn[2C], Moses [3C] and Razavy [4C] that used the Born-Neumann power seriessolution of the acoustic Lippmann-Schwinger equation, and a concomitantexpansion of the interaction in “orders-of-the-data”. Reversion of theBorn-Neumann series leads to an order-by-order scheme for evaluating theterms of the series representation of the scattering interaction interms of the measured data; e.g., only the on-shell reflection amplitudeis required to invert for a local interaction. In principle, the methodis completely general and requires no prior information about the targetor the propagation details of the probe signal within the target.

The only fundamental limitation of the approach appears to be the finiteradius of convergence of the Born-Neumann series solution of theacoustic Lippmann-Schwinger equation. This is generally analyzed usingthe “spectral radius” of the Fredholm kernel of this equation [Morse andFeshbah, [5C]; Newton, [6C]], and in particular by the L²-norm of thiskernel. References and very clear discussions of the issues involved inthe convergence of the Born-Neumann forward scattering series can befound in [Goldberger and Watson [7C]; Newton [6C]]. Despite thislimitation, Weglein and co-workers [1] have made significant progressusing this approach by introducing the idea of “subseries” within theBorn-Neumann expansion, which are associated with specific inversiontasks. This expresses the inversion series in terms of a set of subtaskswhich can be carried out separately from one another.

A particularly significant benefit of this approach is the fat that theconvergence properties of the subseries studied to date are much morefavorable than those of the full Born-Neumann series. Indeed, empiricalevidence has been very encouraging regarding the convergence of theinverse series. However, the nature of the kernel of theLippmann-Schwinger equation, viewed as an equation for the interactionin terms of the T-operator, is such that its maximum eigenvalue alwaysdepends on the explicit nature of the on- and off-shell T-matrix andgeneral statements regarding convergence are difficult to obtain[Prosser [8C]].

Another, more robust approach to solving integral equations is that dueto Fredholm [9C], which can be viewed as a generalization of thewell-known Cramer's method for solving systems of linear simultaneousalgebraic equations. Consequently, fundamental to the approach is acontinuous generalization of the determinant of coefficients and itsminors. Under the circumstances that the integral equation is of theVolterra type, the “Fredholm determinant” can be shown to equal one andthe Fredholm solution reduces to a Born-Neumann expansion, all-be-it onethat converges absolutely independent of the scattering interactionstrength. Consequently, for such Volterra equations, the Born-Neumannexpansion possesses the most robust convergence properties for which onecan hope.

Some years ago, Sams and Kouri [10C] (for noniterative computations inquantum scattering) and Kouri [11C] (for electromagnetic scattering)showed that one could carry out a renormalization transformation of theLippmann-Schwinger equation into a Volterra equation form. Although theVolterra equations for quantum scattering were well known [Goldbergerand Watson [7C]; Newton [6C]], previous studies had focused almostexclusively on their use for studying the analytic structure of theS-matrix and the scattering state. The work of Kouri and co-workersconcentrated on making use of the Volterra form of the scatteringequations to create a noniterative computational algorithm. Theirapproach, however, made essential use of the “triangular” character ofthe Volterra equation kernel, which in one dimension (1D) isK(z, z′)=0, z≧z′ or K(z, z′)=0; z≦z′  (165)combined with a Newton-Cotes quadrature to solve the equations by anoniterative recursion. However, it is also well-known that theproperty, Equation (165), underlies the extremely robust nature of theconvergence of these Volterra equations with respect to an interativesolution [Morse and Feshbah [5C]; Newton [6C]]. Indeed, the Born-Neumannseries solution of the Volterra equation converges absolutely,irrespective of the magnitude of the (in general complex) couplingstrength of the interaction! Furthermore, the convergence depends on theglobal behavior of the interaction (essentially whether it is measurablein a particular sense) and not on its smoothness. For 1D interactionshaving compact support (and for even more general interactions in thecase of 3D scattering), the iterative solution of the Volterra equationconverges uniformly on any closed domain of definition in the scatteringposition variable. Again, under certain relatively weak conditions onthe interaction, the iterative solution is an entire function of thescattering wave number, k [Newton [6C]].

Thus, the possible benefits of formulating acoustic scattering in termsof Volterra kernels appear substantial. The infinitely large radius ofconvergence of the Born-Neumann series solution of the Volterra equationis of especial interest from the standpoint of the inverse acousticscattering approach of Weglein and co-workers [1C]. It seems natural,therefore, to investigate possible benefits of using the renormalizationtechnique as a framework for developing an inverse scattering series. Infact, we shall show that it is possible to establish general, rigorousconvergence properties for the inverse acoustic scattering series forthe first time, and in the process show that its radius of convergenceis also infinite! We shall restrict our discussion here to 1D scatteringbut our approach is completely general and extends to higher dimensions[12C].

This portion of this application is organized as follows. In SectionXIII, we discuss renormalization of the Lippmann-Schwinger equation foracoustic scattering and introduce an auxiliary transition operator,{tilde over (T)}. This is used as the framework to analyze theconvergence of the forward scattering Born-Neumann series. The approachis illustrated by applying it to scattering by a Dirac delta functionmodel interaction. In Section XIV, we show the relationship between theinteraction as a function of the physical T-operator and as a functionof the auxiliary {tilde over (T)}-operator. We next analyze thenon-local nature of {tilde over (T)} in the coordinate representation,and then use the results to establish the convergence properties of theVolterra-based Born-Neumann inverse series for the interaction. Weinclude in this Section an application to the Dirac delta functioninteraction. Next, in Section XV, the Volterra inverse series is appliedto the case of sound scattering by either a square well or barrier. Ourconclusions are given in Section XVI.

XIII. Renormalization of the Lippmann-Schwinger Equation

A. Derivation of the Renormalization Transformation and AuxiliaryTransition Operator {tilde over (T)}

We assume that the reader is familiar with the acoustic scatteringLippmann-Schwinger equation for the transition operator, T, given by[Razavy [4C]; Goldberger and Watson [5C]; Newton [6C]]

$\begin{matrix}{T = {{\gamma\; V} + {\gamma\;{VG}_{0k}^{+}T}}} & (166)\end{matrix}$where G_(0k) ⁺ is the causal free Green's operator, multiplied by afactor of k²,

$\begin{matrix}{G_{0k}^{+} = \frac{k^{2}}{E - H_{0} + {{\mathbb{i}}\; ɛ}}} & (167)\end{matrix}$k²=E (i.e., k is the frequency associated with the incident acousticwave), H₀ governs the “free propagation” of the acoustic wave, and ΔV isthe interaction responsible for the scattering, with γ being thecoupling parameter characterizing the strength of the interaction. Ingeneral, γ is complex. The additional factor of k² results from the factthat in acoustic scattering (as in general for scattering governed by aHelmholtz type wave equation), the interaction responsible forscattering depends on k². The full acoustic wave propagation (scatteringprocess) is thus governed by the operator H,H=H ₀ +k ² γV  (168)The present 1D acoustic scattering problem in the coordinaterepresentation leads to

$\begin{matrix}\begin{matrix}{{T\left( {z,z^{\prime}} \right)} = {{\gamma\;{V\left( {z,z^{\prime}} \right)}} +}} \\{\int_{- \infty}^{+ \infty}{{\mathbb{d}z^{''}}\gamma\;{V\left( {z,z^{''}} \right)}{\int_{- \infty}^{+ \infty}{{\mathbb{d}z^{\prime''}}{G_{0k}^{+}\left( {z^{''},z^{\prime''}} \right)}{T\left( {z^{\prime''},z^{\prime}} \right)}}}}}\end{matrix} & (169)\end{matrix}$By incorporating this factor of k² into the Green's function, we areable to treat the remaining portion of the interaction that dependspurely on the spatial variation of the scattering interaction.Initially, we restrict ourselves to “local scattering media”, so thatV(z, z′)=V(z)δ(z−z′) and therefore

$\begin{matrix}\begin{matrix}{{T\left( {z,z^{\prime}} \right)} = {{\gamma\;{V(z)}{\delta\left( {z - z^{\prime}} \right)}} +}} \\{\gamma\;{V(z)}{\int_{- \infty}^{+ \infty}{{\mathbb{d}z^{''}}{G_{0k}^{+}\left( {z,z^{''}} \right)}{T\left( {z^{''},z^{\prime}} \right)}}}}\end{matrix} & (170)\end{matrix}$

The non-local character of the causal free Green's function,

G_(0k)⁺(z, z^(″)),reflected in its not commuting with γV, is responsible for the fact thatT(z, z″) is also generally non-local; i.e., it is never diagonal in thecoordinate representation (except for a local, Dirac delta functioninteraction, V(z, z′)=V(z)δ(z−z′)=λδ(z−z′)δ(z−z₀)). For 1D causalscattering boundary conditions,

G_(0k)⁺(z, z^(″))is explicitly

$\begin{matrix}{{G_{0k}^{+}\left( {z,z^{''}} \right)} = {{- \frac{{\mathbb{i}}\; k}{2}}{\mathbb{e}}^{{\mathbb{i}}\; k{{z - z^{''}}}}}} & (171)\end{matrix}$The general scattering amplitude is determined by the matrix elements ofthe T-operator, usually computed in the momentum representation, T(k′,k″), given byT(k′, k″)=<k′|T|k″)  (172)where in general, k′, k″ and the on-energy-shell wave number, k=√{squareroot over (E)} need not be equal to one another. The physical“reflection scattering amplitude”, denoted r(k), results when|k′|=|k″|=|k| and k′=−k:r(k)=(−ikπ)(−k|T|k>  (173)In 1D scattering, one can also identify the transmission amplitude,t(k), given byt(k)=1+(−ikπ)<k|T|k>  (174)

In the work of Sams and Kouri [10C], the renormalization transformationto a Volterra equation results from eliminating the |z−z″|-argument inthe free Green's function in Equation (170). This is done by dividingthe integration over z″ into segments from −∞ to z and from z to ∞:

$\begin{matrix}\begin{matrix}{{T\left( {z,z^{\prime}} \right)} = {{\gamma\;{V(z)}{\delta\left( {z - z^{\prime}} \right)}} -}} \\{{\frac{{\mathbb{i}}\; k}{2}\gamma\;{V(z)}{\int_{- \infty}^{z}{{\mathbb{d}z^{''}}{\mathbb{e}}^{{\mathbb{i}}\;{k{({z - z^{''}})}}}{T\left( {z^{''} - z^{\prime}} \right)}}}} -} \\{\frac{{\mathbb{i}}\; k}{2}\gamma\;{V(z)}{\int_{z}^{+ \infty}{{\mathbb{d}z^{''}}{\mathbb{e}}^{{- {\mathbb{i}}}\;{k{({z - z^{''}})}}}{T\left( {z^{''},z^{\prime}} \right)}}}}\end{matrix} & (175)\end{matrix}$One then adds and subtracts

${{- \frac{{\mathbb{i}}\; k}{2}}\gamma\;{V(z)}{\int_{z}^{- \infty}{{\mathbb{d}z^{''}}{\mathbb{e}}^{{\mathbb{i}}\;{k{({z - z^{''}})}}}{T\left( {z^{''},z^{\prime}} \right)}}}},$and after simple manipulation, oneobtains

$\begin{matrix}\begin{matrix}{{T\left( {z,z^{\prime}} \right)} = {{\gamma\; V\;{(z)\left\lbrack {{\delta\left( {z - z^{\prime}} \right)} - {\frac{{\mathbb{i}}\; k}{2}{\mathbb{e}}^{{\mathbb{i}}\;{kz}}{\int_{- \infty}^{+ \infty}{{\mathbb{d}z^{''}}{\mathbb{e}}^{{- {\mathbb{i}}}\;{kz}^{''}}{T\left( {z^{''},z^{\prime}} \right)}}}}} \right\rbrack}} -}} \\{\frac{{\mathbb{i}}\; k}{2}\gamma\;{V(z)}{\int_{z}^{+ \infty}{{\mathbb{d}{Z^{''}\left\lbrack {{\mathbb{e}}^{{- {\mathbb{i}}}\;{k{({z - z^{''}})}}} - {\mathbb{e}}^{{\mathbb{i}}\;{k{({z - z^{''}})}}}} \right\rbrack}}{T\left( {z^{''},z^{\prime}} \right)}}}}\end{matrix} & (176)\end{matrix}$It can be verified that this is equivalent to writing

G_(0k)⁺(z, z^(″))as

$\begin{matrix}{{G_{0k}^{+}\left( {z,z^{''}} \right)} = {{{\overset{\sim}{G}}_{0k}\left( {z,z^{''}} \right)} - {\frac{{\mathbb{i}}\; k}{2}{\mathbb{e}}^{{\mathbb{i}}\;{k{({z - z^{''}})}}}}}} & (177)\end{matrix}$so that

$\begin{matrix}\begin{matrix}{{{\overset{\sim}{G}}_{0k}\left( {z,z^{''}} \right)} = {- {\frac{{\mathbb{i}}\; k}{2}\left\lbrack {{\mathbb{e}}^{{\mathbb{i}}\;{k{({z^{''} - z})}}} - {\mathbb{e}}^{{- {\mathbb{i}}}\;{k{({z^{''} - z})}}}} \right\rbrack}}} \\{{\equiv {k\;{\sin\left\lbrack {k\left( {z^{''} - z} \right)} \right\rbrack}}},{z < z^{''}}}\end{matrix} & (178) \\{{\text{~~~~~~~~~~~~~~~~} = 0},{z \geq z^{''}}} & (179)\end{matrix}$In abstract operator notation, this is

$\begin{matrix}{G_{0k}^{+} = {{\overset{\sim}{G}}_{0k} - {{\mathbb{i}}\; k\;\pi\left. k \right\rangle\left\langle k \right.}}} & (180)\end{matrix}$

This relation is extremely useful in our subsequent analysis and weshall make much use of it. Notice that the Green's operator {tilde over(G)}_(0k) differs from the usual causal one,

G_(0k)⁺,by a solution of the homogeneous equation [Newton, [6C]]:

$\begin{matrix}{{\left( {E - H_{0}} \right)G_{0k}^{+}} = k^{2}} & (181)\end{matrix}$(E−H ₀){tilde over (G)} _(0k) =k ²  (182)(E−H ₀) [−ikπ|k><k|]=[−ikπ|k><k|(E−H ₀)=0  (183)The abstract version of Equation (176) results from substitutingEquation (180) into Equation (166):T=γV[1−ikπ|k><k|T]+γV{tilde over (G)} _(0k) T  (184)

Next we note that the action of T on the initial state |k> is of theformT|k>=γV[1−ikπ<k|T|k>]|k>+γ{tilde over (G)} _(0k) T|k>  (185)(unknown) constant, c_(k), asc _(k)=1−ikπ≦k|T|k≧=≡t(k)  (186)we see thatT|k>=γVc _(k) |k>+γV{tilde over (G)} _(0k) T|k>  (187)The relationship between T|k> and the Lippmann-Schwinger pressure state,

P_(k)⁺⟩,is

$\begin{matrix}{{\sqrt{2\pi}T\left. k \right\rangle} = {\gamma\; V\left. P_{k}^{+} \right\rangle}} & (188)\end{matrix}$and thus

$\begin{matrix}{\left. P_{k}^{+} \right\rangle = {{\sqrt{2\pi}c_{k}\left. k \right\rangle} + {{\overset{\sim}{G}}_{0k}\gamma\; V\left. P_{k}^{+} \right\rangle}}} & (189)\end{matrix}$Clearly, the factor C_(k) is simply a normalization constant and one candefine an auxiliary pressure state vector |p_(k)>, in relation to

P_(k)⁺⟩,according to|P _(k) ⁺ >=c _(k) |p _(k>)  (190)|p _(k)>=√{square root over (2π)}|k>+{tilde over (G)}_(0k) γV|p_(k>)  (191)

The coordinate representation, <z|p_(k)>=p_(k)(z) satisfies

$\begin{matrix}{{p_{k}(z)} = {{\mathbb{e}}^{{\mathbb{i}}\;{kz}} + {k{\int_{z}^{+ \infty}{{\mathbb{d}z^{''}}{\sin\left\lbrack {k\left( {z^{''} - z} \right)} \right\rbrack}\gamma\;{V\left( z^{''} \right)}{p_{k}\left( z^{''} \right)}}}}}} & (192)\end{matrix}$which is recognized as an inhomogeneous Volterra integral equation ofthe second kind. We remark here that Volterra equations involvingimproper limits (i.e., ±∞) still converge absolutely for |γ|<∞, but theymust satisfy additional restrictions on the z-dependence of theinteraction. This is especially true in order for their iterativesolutions to converge uniformly on any closed interval [z₁, z₂]. It issufficient that the interaction V(z) have compact support and |V(z)| bemeasureable. It remains true even for infinite ranged interactions solong as they decay sufficiently rapidly and are not too singular. Thisis discussed for similar Volterra equations in [Goldberger and Watson[5C]; Newton [6C]]. Throughout our discussion, we assume that suchconditions are met. By Equation (190), Lp*) results from renormalizing

P_(k)⁺⟩according to

$\begin{matrix}{\left. p_{k} \right\rangle = \frac{\left. P_{k}^{+} \right\rangle}{c_{k}}} & (193)\end{matrix}$

In fact, c_(k) is essentially the inverse of the Jost function [Newton[6C]]. We remark that the above expression also provides the physialinterpretation of the “Volterra pressure wave”, p_(k)(z) [13C]. Clearly,it represents a wave produed by an incident plane wave having anamplitude equal to 1/c_(k)≡1/t(k). This leads to a reeted wave with theamplitude r(k)|t(k) and a transmitted wave with amplitude exatly equalto one. Of course, such an incident wave cannot, in general, be createdexperimentally since it requires advance knowledge of the effect of thescatterer in the form of 1/t(k). However, this does not alter theinterpretation of the wave p_(k)(Z).

Let us now return to Equation (184), and define an auxilliary transitionoperator, {tilde over (T)}, according toT={tilde over (T)}[1−ikπ|k><k|T]  (194)It can be verified that{tilde over (T)}=γV+γV{tilde over (G)} _(0k) {tilde over (T)}  (195)and this is the fundamental equation which will be used to analyze theinverse series for γV. (Note that the operator inverse,[1−ikπ|k><k|T]⁻¹, should always exist. This essentially requires thatthe operator ikπ|k><k|T not have any eigenvalues equal to +1. Aworst-case would correspond to the inverse of T being equal toikπ|k><k|, which cannot occur since T does not commute with H₀ while|k><k| does.) It is instructive to evaluate explicitly the normalizationconstant, c_(k), in terms of the solution of the Volterra equation. Thiscan be quite done by combining Equations (185), (186) and (194) to writeT|k>=c _(k) {tilde over (T)}|k>  (196)expressed asc _(k)=1−ikπ<k|{tilde over (T)}|k>c _(k)  (197)so that

$\begin{matrix}{c_{k} = \frac{1}{1 + {{\mathbb{i}}\; k\;\pi\left\langle {k{\overset{\sim}{T}}k} \right\rangle}}} & (198)\end{matrix}$Thus, the renormalized or auxiliary pressure state, |p_(k)>, is given by

$\begin{matrix}{\left. p_{k} \right\rangle = {\left. P_{k}^{+} \right\rangle\left\lbrack {1 - {{\mathbb{i}}\; k\mspace{2mu}\pi\left\langle {k{\overset{\sim}{T}}k} \right\rangle}} \right\rbrack}} & (199)\end{matrix}$

The physical reflection amplitude, r(k), is given byr(k)=−ikπ<−k|T|k>=−t(k)ikπ<−k|{tilde over (T)}|k>  (200)These relations provide us with the necessary tools to expressauxilliary amplitudes in terms of the physical amplitudes.

B. Convergence of the Born-Neumann Series for |p_(k)> and {tilde over(T)}

On one hand, the convergence of the Born-Neumann series for either

P_(k)⁺⟩or T is well-known to depend critically on the size of the couplingconstant, γ(or equivalently, on the size of the “contrast” between thepropagation under H₀ and that under H=H₀+k²γP) [Goldberger and Watson[5]; Newton [6]]. On the other hand, it is also well-known thatiterative solutions of either Equation (192) or (195) convergeabsolutely for |γ|∞[Newton [6]]. Furthermore, the iteration of Equation(192) converges uniformly on any closed domain of z (for a wide class ofinteractions). It is useful to stress the origin of this robustnesssince it turns out to be the basis of the convergence of theVolterra-based inverse series for γV. The kernel of Equation (192) canbe written (for all z, z″) as

$\begin{matrix}{{{\gamma\;{K\left( {z,z^{''}} \right)}} = {k\;\gamma\;{\sin\left\lbrack {k\left( {z^{''} - z} \right)} \right\rbrack}{V\left( z^{''} \right)}}},{z < z^{''}}} & (201) \\{{\text{~~~~~~~~~~~~~~~} \equiv 0},{z \geq z^{''}}} & (202)\end{matrix}$

According to the discussion in [Newton [6]; see also those in Rodbergand Thaler [14] and Mathews and Walker [15]] one characterizes theconvergence in terms of Fredholm's method of solution. This method isthe continuum analogue of solving a linear system of algebraicequations, and it expresses the inverse of the integral kernel in termsof the ratio of the first Fredholm minor to the Fredholm determinant, Δ.The determinant Δ can be expressed as an infinite series of the form(for the acoustic case)

$\begin{matrix}{\Delta = {\sum\limits_{n = 0}^{\infty}{(\gamma)^{n}\kappa_{n}}}} & (203)\end{matrix}$whereκ_(n) ≡T _(r)(K″)  (204)It is not difficult to verify that for the Volterra kernel, K(z, z″),above,κ_(n)=δ_(n0)  (205)and consequently, for such kernels,Δ≡1  (206)regardless of the strength of the scatterer, γ. Furthermore, by use ofHadamard's theorem [6C, 14C], it then can be proved that the infiniteseries for the first Fredholm minor converges absolutely and uniformlyfor γ in the entire complex plane (more details are given in theAppendix C). It also has been established [Mathews and Walker [15C]]that when the Fredholm determinant equals one, the Fredholm solution isidentical to the Born-Neumann iterative solution of the integralequation. We conclude that iterative solutions of Volterra integralequations possess the most robust convergence possible. While it is truethat these convergence properties are independent of the strength of γ,there are conditions on the analytical structure allowed for thescattering interaction. These have to do with the integrability of anysingularities and the behavior at infinity. They are discussed in[Newton [6C]] in some detail. If the interaction has compact support,and is not too singular, then the convergence is of the strongestcharacter (i.e., absolute and uniform, leading to entire functions ofwave number k and coupling γ).

It is therefore clear that the essential property of the Volterra kernelis that it satisfies Equations (201)–(202); as noted in [Newton [6C]],this is the continuous version of the “triangular” property of matries.It is equivalent to the property that the Fredholm determinant ofEquation (192) or (195) is identially one. Furthermore, it ensures thatthe Born-Neumann series for p_(k)(Z), obtained from Equation (195), isuniformly convergent on any closed domain of z, for a wide class ofinteractions.

We stress that this is all well-known. We have included it explicitlyhere because its implications for the inverse scattering seriesdetermining γV have never been explicated. In addition, it is perhapsnot appreciated that the original Lippmann-Schwinger equation itself canbe directly iterated in a fashion that is also everywhere absolutelyconvergent! Obviously, such an iteration must differ from thestraightforward iteration of the Lippmann-Schwinger equation,

${\left. P_{k}^{+} \right\rangle = {{\sqrt{2\pi}\left. k \right\rangle} + {G_{0k}^{+}\gamma\; V\left. P_{k}^{+} \right\rangle}}},$which leads to

$\begin{matrix}{\left. P_{k}^{+} \right\rangle = {\sqrt{2\pi}{\sum\limits_{n = 0}^{\infty}{\left( {G_{0k}^{+}\gamma\; V} \right)^{n}\left. k \right\rangle}}}} & (207)\end{matrix}$the proof of whose convergence depends on the L²-norm of the kernel,

G_(0k)⁺γ V₂.In fact, wean simply iterate Equation (189) for the Lippmann-Schwingerstate

P_(k)⁺⟩:

$\begin{matrix}{\left. P_{k}^{+} \right\rangle = {{\sqrt{2\pi}{\sum\limits_{n = 0}^{\infty}{\left( {{\overset{\sim}{G}}_{0k}\gamma\; V} \right)^{n}c_{k}\left. k \right\rangle}}} = {\sqrt{2\pi}c_{k}{\sum\limits_{n = 0}^{\infty}{\left( {{\overset{\sim}{G}}_{0k}\gamma\; V} \right)^{n}\left. k \right\rangle}}}}} & (208)\end{matrix}$

We stress that even though c_(k) is unknown in Equation (208), it issimply a number and can be calculated directly from the knowniterate-vectors,

(G_(0k)⁺γ v)^(n)k⟩.Thus,

$\begin{matrix}{c_{k} = {\frac{1}{1 + {{\mathbb{i}}\; k\;\pi{\sum\limits_{n = 0}^{\infty}{\left\langle k \right.\gamma\;{V\left( {{\overset{\sim}{G}}_{0k}\gamma\; V} \right)}^{n}\left. k \right\rangle}}}} = {t(k)}}} & (209)\end{matrix}$This is equivalent to the iterative solution for |p_(k)> but the pointwe wish to stress is that the standard, physical Lippmann-Schwingerequation an be iterated in an absolutely convergent fashion, independentof the strength of the interaction. Ofourse, this is simply a reflectionof the fact that the Lippmann-Schwinger equation is neither purely aFredholm or Volterra equation. Therefore, it can manifest theconvergence characteristics of either, depending on the manner in whichit is written and iterated.

C. Illustrative Example

It is helpful toonsider an example problem in order to appreiate betterthe vast difference in convergence between the Born-Neumann series basedon the Lippmann-Schwinger Fredholm equation and the renormalizedLippmann-Schwinger Volterra equation. A convenient and simple modelscattering interaction is the Dirac delta function:γV(z)=γδ(z−z ₀)  (210)The solution to the Lippmann-Schwinger equation is found from notingthat

$\begin{matrix}{{P_{k}^{+}(z)} = {{\mathbb{e}}^{{\mathbb{i}}\;{kz}} - {\frac{{\mathbb{i}}\; k\;\gamma}{2}{\mathbb{e}}^{{\mathbb{i}}\; k{{z - z_{0}}}}{P_{k}^{+}\left( z_{0} \right)}}}} & (211)\end{matrix}$This implies that

$\begin{matrix}{{P_{k}^{+}\left( z_{0} \right)} = \frac{{\mathbb{e}}^{{\mathbb{i}}\;{kz}_{0}}}{\left( {1 + \frac{{\mathbb{i}}\; k\;\gamma}{2}} \right)}} & (212)\end{matrix}$so the exact solution is

$\begin{matrix}{{P_{k}^{+}(z)} = {{\mathbb{e}}^{{\mathbb{i}}\;{kz}} - {\frac{\frac{{\mathbb{i}}\; k\;\gamma}{2}}{\left( {1 + \frac{{\mathbb{i}}\; k\;\gamma}{2}} \right)}{\mathbb{e}}^{{\mathbb{i}}\;{kz}_{0}}{\mathbb{e}}^{{\mathbb{i}}\; k{{z - z_{0}}}}}}} & (213)\end{matrix}$The Born-Neumann series solution is given by

$\begin{matrix}{{P_{k}^{+}(z)} = {{\mathbb{e}}^{{\mathbb{i}}\;{kz}} - {\left( \frac{{\mathbb{i}}\; k\;\gamma}{2} \right){\mathbb{e}}^{{\mathbb{i}}\;{kz}_{0}}{{\mathbb{e}}^{{\mathbb{i}}\; k{{z - z_{0}}}}\left\lbrack {1 - \frac{{\mathbb{i}}\; k\;\gamma}{2} + \left( \frac{{\mathbb{i}}\; k\;\gamma}{2} \right)^{2} + \cdots} \right\rbrack}}}} & (214)\end{matrix}$It is clear that the convergence of this series is determined by therequirement |kγ/2|<1, which is just the condition for the convergence ofa power series expansion of (1+ikγ/2)⁻¹. It is also evident that theBorn-Neumann series is convergent only at low energies in this casesince k=√{square root over (E)}, and therefore only for sufficiently lowE will th e convergence condition will be satisfied. This is theopposite of the usual situation that applies to quantum scattering[Goldberger and Watson [5C]]. Of course, in this simple example, one canrecognize that the series can be analytically summed to yield the exactresult valid at all k and γ. In general, that will not be the case.

We next consider the Born-Neumann series for p_(k)(z); Equation (192)then becomes

$\begin{matrix}\begin{matrix}{{p_{k}(z)} = {{\mathbb{e}}^{{\mathbb{i}}\;{kz}} + {k{\int_{z}^{+ \infty}{{\mathbb{d}z^{''}}{\sin\left\lbrack {k\left( {z^{''} - z} \right)} \right\rbrack}{{\gamma\delta}\left( {z^{''} - z_{0}} \right)}{\mathbb{e}}^{{\mathbb{i}}\;{kz}^{''}}}}} +}} \\{k^{2}{\int_{z}^{+ \infty}{{\mathbb{d}z^{''}}{\sin\left\lbrack {k\left( {z^{''} - z} \right)} \right\rbrack}\gamma\;{\delta\left( {z^{''} - z_{0}} \right)}}}} \\{{\int_{z^{''}}^{+ \infty}{{\mathbb{d}z^{\prime''}}{\sin\left\lbrack {k\left( {z^{\prime''} - z^{''}} \right)} \right\rbrack}{{\gamma\delta}\left( {z^{\prime''} - z_{0}} \right)}{\mathbb{e}}^{{\mathbb{i}}\;{kz}^{\prime''}}}} + \cdots}\end{matrix} & (215)\end{matrix}$We see that all terms higher than first order in the interaction vanishidentically due to the appearance of the factor sin[k(z₀−z₀)]. Thus, forz≧z₀, we obtain exactlyp_(k)(z)=e ^(ikz)  (216)and for z<z₀, we obtain exactlyp _(k)(z)=e ^(ikz) +kγ sin[k(z ₀ −z)]e ^(ikz) ⁰   (217)

This does not complete the analysis since we must also evaluate c_(k)using only information generated by the iterative solution for p_(k)(Z).This is simple using Equations (196)–(200) and yields

$\begin{matrix}{c_{k} = \frac{1}{1 + {\frac{{\mathbb{i}}\; k\;\gamma}{2}{\int_{- \infty}^{+ \infty}{{\mathbb{d}z}\;{\mathbb{e}}^{{- {\mathbb{i}}}\;{kz}}{\delta\left( {z - z_{0}} \right)}{p_{k}(z)}}}}}} & (218) \\{\text{~~~} = \frac{1}{1 + {\frac{{\mathbb{i}}\; k\;\gamma}{2}{\int_{- \infty}^{+ \infty}{{\mathbb{e}}^{{- {\mathbb{i}}}\;{kz}_{0}}{p_{k}\left( z_{0} \right)}}}}}} & (219)\end{matrix}$

But by Equation (216), this gives

$\begin{matrix}{c_{k} = \frac{1}{1 + \frac{{\mathbb{i}}\; k\;\gamma}{2}}} & (220)\end{matrix}$Therefore the Born-Neumann series for

P_(k)⁺(z)based on the renormalized Lippmann-Schwinger equation results in

$\begin{matrix}{{{P_{k}^{+}(z)} = \frac{{\mathbb{e}}^{{\mathbb{i}}\;{kz}_{0}}}{\left( {1 + \frac{{\mathbb{i}}\; k\;{\gamma/2}}{2}} \right)}},{z \geq z_{0}}} & (221) \\{{\text{~~~~~~~~} = {{\mathbb{e}}^{{\mathbb{i}}\;{kz}} - {\frac{\frac{{\mathbb{i}}\; k\;\gamma}{2}}{\left( {1 + \frac{{\mathbb{i}}\; k\;\gamma}{2}} \right)}{\mathbb{e}}^{2{\mathbb{i}}\;{kz}_{0}}{\mathbb{e}}^{{- {\mathbb{i}}}\;{kz}}}}},{z < z_{0}}} & (222)\end{matrix}$

We conclude that the Volterra-based iteration converges to the exactanswer with just the first order term, all higher terms being zero. Thefundamental difference between the Volterra and Fredholm iteratedexpressions is that the former does not involve a power series expansionof the normalization factor, c_(k), whose convergence would haverequired that |kγ/2| be less than one. Instead, c_(k) has been factoredout by renormalizing from

P_(k)⁺(z)to p_(k)(z). We emphasize that this renormalization follows for anyscattering problem that is expressible in terms of Green's functions

G_(0k)^(∓),since it is true in general (for 1D scattering) that

$\begin{matrix}{G_{0k}^{\pm} = {{\overset{\sim}{G}}_{0k} \mp {{\mathbb{i}}\; k\;\pi\left. k \right\rangle\left\langle k \right.}}} & (223)\end{matrix}$We also note that analogous relationships have been derived for 3Dscattering Green's functions. We now turn to consider the inversescattering series for the interaction, γV.XIV. The Inverse Scattering Series for γV

A. Fredholm and Volterra Born-Neumann Series for γV

We begin by establishing that distinct Born-Neumann series for γV can beobtained from Equation (166) for T or Equation (195) for T. We solveEquation (166) for γV as

$\begin{matrix}{{\gamma\; V} = {T\left( {1 + {G_{0k}^{+}T}} \right)}^{- 1}} & (224) \\{\text{~~~~~} = {T\left( {1 + {{\overset{\sim}{G}}_{0k}T} - {{\mathbb{i}}\; k\;\pi\left. k \right\rangle\left\langle k \right.T}} \right)}^{- 1}} & (225)\end{matrix}$But Equation (194), this yields

$\begin{matrix}{{\gamma\; V} = {T\left( {\left\lbrack {1 - {{\mathbb{i}}\; k\;\pi\left. k \right\rangle\left\langle k \right.T}} \right\rbrack + {{\overset{\sim}{G}}_{0k}{\overset{\sim}{T}\left\lbrack {1 - {{\mathbb{i}}\; k\;\pi\left. k \right\rangle\left\langle k \right.T}} \right\rbrack}}} \right)}^{- 1}} & (226) \\{\text{~~~~~} = {{T\left\lbrack {1 - {{\mathbb{i}}\; k\;\pi\left. k \right\rangle\left\langle k \right.T}} \right\rbrack}^{- 1}\left( {1 + {{\overset{\sim}{G}}_{0k}\overset{\sim}{T}}} \right)^{- 1}}} & (227)\end{matrix}$so that finally,γV={tilde over (T)}(1+{tilde over (G)} _(0k) {tilde over (T)})⁻¹  (228)It follows that, provided they converge, γV can be obtained from eitherof the following Born-Neumann series expansions:

$\begin{matrix}{{\gamma\; V} = {\sum\limits_{n = 0}^{\infty}{T\left( {G_{0k}^{+}T} \right)}^{n}}} & (229)\end{matrix}$and

$\begin{matrix}{{\gamma\; V} = {\sum\limits_{n = 0}^{\infty}{\overset{\sim}{T}\left( {{- {\overset{\sim}{G}}_{ok}}\overset{\sim}{T}} \right)}^{n}}} & (230)\end{matrix}$

The convergence properties of Equation (229) depend, of course, on thespectral radius of the kernel

G_(0k)⁺T,which in turn depends crucially on both the on- and off-shell elementsof the T-matrix. For this reason, general conclusions regarding theconvergence of Equation (229) have been extremely difficult to obtaindespite heroic efforts [Prosser [8C]]. We shall see that this is not thecase for Equation (230)!

The convergence properties of Equation (230) will be studied using theFredholm method of solving Equation (228). To do so requires knowledgeof the properties of the kernel {tilde over (G)}_(0k){tilde over (T)},which are yet to be established. It is clear, however, that when bothexpansions converge, they must agree, since convergent power seriesyield a unique result [Kaplan [16C]]. In order to investigate theconvergence of Equation (230), we now consider the non-local characterof {tilde over (T)} in the coordinate representation.

B. Non-local Character of {tilde over (T)} (z, z′)

The aim of this subsection is to establish that {tilde over (T)}(z,z′)=0 when z>z′. It is not difficult to show that Equation (195) has thesolution{tilde over (T)}=γV+γV{tilde over (G)}γV  (231)where{tilde over (G)}={tilde over (G)} ₀ +{tilde over (G)} ₀γV{tilde over(G)}  (232)(see R. G. Newton, 1982, pp. 343–344; especially Equation 12.42 and thefollowing unnumbered equation). From Equation 12.40a in Newton, we seethatG ⁺(k;z,z′)=−kψ ⁺(k,z _(<))f(k,z _(>))  (233)where ψ⁺ (k, z) is the regular (physical or causal) scattering solutionof the interacting Shrodinger equation and f(k,z) is an irregularsolution of the same equation, introduced by Jost [6C]. Then defining aninteracting Green's function, {tilde over (G)}(k; z, z′), that vanishesfor z≧z′, it is easy to see that{tilde over (G)}(k;z,z′)=G ⁺(k;z,z′)+kψ ⁺(k,z′)f(k,z)  (234)

Obviously, kψ⁺(k,z′)f (k,z) satisfies the homogeneousinteracting-Green's function Schrodinger equation. Then{tilde over (G)}(z,z′)=0, z≧z′  (235)and for local potentials

$\begin{matrix}\begin{matrix}{{\overset{\sim}{T}\left( {z,z^{\prime}} \right)} = {{\gamma\;{V(z)}{\delta\left( {z - z^{\prime}} \right)}} +}} \\{\gamma^{2}{\int_{- \infty}^{+ \infty}{{\mathbb{d}z^{''}}{\int_{- \infty}^{+ \infty}{{\mathbb{d}z^{\prime\prime\prime}}{V(z)}{\delta\left( {z - z^{''}} \right)}}}}}} \\{{\overset{\sim}{G}\left( {z^{''},z^{\prime\prime\prime}} \right)}{V\left( z^{\prime\prime\prime} \right)}{\delta\left( {z^{\prime\prime\prime} - z^{\prime}} \right)}}\end{matrix} & (236)\end{matrix}$or{tilde over (T)}(z,z′)=γV(z)δ(z−z′)+γ² V(z){tilde over(G)}(z,z′)V(z′)  (237)It is therefore clear that as a function of either z or z′, {tilde over(T)}(z, z′) has support determined by V(z) or V(z′). Also, for z>z′, thefirst term on the R.S. of Equation (237) is zero due to the Dirac deltafunction and the second term is zero due to {tilde over (G)}(z, z′).Therefore, we have proved that{tilde over (T)}(z,z′)=0, z>z′  (238)Finally, {tilde over (T)}(z, z′) has an inerrable singularity at z=z′.

C. Convergence of the Inverse Series for γV

We note next that the kernel of the Volterra-based Born-Neumann seriesfor γV, Equation (238), is given by{tilde over (K)}(z, z′)=<z|{tilde over (G)} _(0k) {tilde over(T)}|z′)  (239)It is necessary to compute Tr({tilde over (K)}^(n)), but it issufficient to examine Tr({tilde over (K)}²) to see how the general casebehaves:

$\begin{matrix}{{{Tr}\left( {\overset{\sim}{K}}^{2} \right)} = {\int_{- \infty}^{+ \infty}{{\mathbb{d}z}{\int_{- \infty}^{+ \infty}{{\mathbb{d}z^{\prime}}{\overset{\sim}{K}\left( {z,z^{\prime}} \right)}{\overset{\sim}{K}\left( {z^{\prime},z} \right)}}}}}} & (240)\end{matrix}$This can be written as

$\begin{matrix}\begin{matrix}{{{Tr}\left( {\overset{\sim}{K}}^{2} \right)} = {\int_{- \infty}^{+ \infty}{{\mathbb{d}z_{1}}\mspace{14mu}\cdots{\int_{- \infty}^{+ \infty}{{\mathbb{d}z_{4}}{{\overset{\sim}{G}}_{0k}\left( {z_{1},z_{2}} \right)}{\overset{\sim}{T}\left( {z_{2},z_{3}} \right)}}}}}} \\{{{\overset{\sim}{G}}_{0k}\left( {z_{3},z_{4}} \right)}{\overset{\sim}{T}\left( {z_{4},z_{1}} \right)}}\end{matrix} & (241)\end{matrix}$However, by the Volterra property of {tilde over (G)}_(0k) and {tildeover (T)}, a nonzero contribution can only occur ifz₁>z₄>z₃>z₂>z₁  (242)which clearly is never satisfied. We conclude that the Volterra propertyis satisfied for the product of two (or more) Volterra kernels andTr({tilde over (K)} ²)=Tr({tilde over (K)} ^(n))≡0, n≧2  (243)It is similarly easy to prove that Tr({tilde over (K)})=0.

We therefore conclude that the Fredholm determinant for Equation (228)equals one. This guarantees that, for a not-too-singular, localinteraction having compact support, the Volterra-based inversescattering series converges absolutely and uniformly independent of thestrength of the interaction! This is an amazing result since it ensuresthat this inverse scattering series always converges for any magnitude(complex) coupling constant.

D. Utilization of the Volterra Inverse Series for γV in Orders of {tildeover (T)} and the Relation to Data Requirements

In order to use the new Volterra inverse series to determine γV, thefinal step is to develop explicit expressions for it in terms of“far-field” measured quantities. The standard Born-Neumann inversion ofthe Lippmann-Schwinger-based approach, to obtain a local potential,requires knowledge only of the reflection amplitude, r(k), as a functionof k. We shall see that additional data are required in order to use theVolterra inverse series. Recall that by Equation (230),

$\begin{matrix}{{\gamma\; V} = {{\sum\limits_{n = 0}^{\infty}\overset{\sim}{T}} - \left( {{\overset{\sim}{G}}_{ok}\overset{\sim}{T}} \right)^{n}}} & (244)\end{matrix}$whereT={tilde over (T)}[1−ikπ|k><k|T]  (245)We shall express γV as a power series in orders of {tilde over (T)}:

$\begin{matrix}{{\gamma\; V} = {\sum\limits_{n = 0}^{\infty}{{\overset{\sim}{\lambda}}^{j}{\overset{\sim}{V}}_{j}}}} & (246)\end{matrix}$where{tilde over (V)} _(j) ={tilde over (T)}(−{tilde over (G)} _(0k) {tildeover (T)}) ^(j−1) , j=1,2,  (247)Next, recall that by Equation (195),

$\begin{matrix}{\overset{\sim}{T} = {\sum\limits_{n = 0}^{\infty}{\left( {\gamma\; V\;{\overset{\sim}{G}}_{ok}} \right)^{n}\gamma\; V}}} & (248) \\{{\overset{\sim}{\lambda}\overset{\sim}{T}} = {\sum\limits_{n = 0}^{\infty}{\left( {\sum\limits_{j = 1}^{\infty}{{\overset{\sim}{\lambda}}^{j}{\overset{\sim}{V}}_{j}{\overset{\sim}{G}}_{ok}}} \right)^{n}{\sum\limits_{j^{\prime} = 1}^{\infty}{{\overset{\sim}{\lambda}}^{j^{\prime}}{\overset{\sim}{V}}_{j^{\prime}}}}}}} & (249)\end{matrix}$since {tilde over (T)} is first order in {tilde over (λ)}. We thencollect coefficients of each power of {tilde over (λ)}^(j):{tilde over (λ)}¹: {tilde over (T)}={tilde over (V)}₁  (250){tilde over (λ)}²: 0={tilde over (V)} ₂ +{tilde over (V)} ₁ {tilde over(G)} _(0k) {tilde over (V)} ₁  (251){tilde over (λ)}³: 0={tilde over (V)} ₃ +{tilde over (V)} ₂ {tilde over(G)} _(0k) {tilde over (V)} ₁ +{tilde over (V)} ₁ {tilde over (G)} _(0k){tilde over (V)} ₁ {tilde over (G)} _(0k) {tilde over (V)} ₁  (252)etc. Matrix elements of these expressions are first evaluated in thek-representation and the results subsequently transformed to thez-representation. This is because the starting expression involves thek-representation matrix elements of {tilde over (T)}. However, using thelower order operators, {tilde over (V)}₁, to express {tilde over(V)}_(j) solely in terms of {tilde over (V)}₁ and {tilde over (G)}_(0k),one can find that, in general,{tilde over (V)} _(j) =−{tilde over (V)} _(j−1) {tilde over (G)} _(ok){tilde over (V)} ₁  (253)

This is the most convenient form with which to evaluate the higher ordercorrections. The Volterra-based expressions can be compared to theBorn-Neumann inverse series based on the usual Lippmann-Schwingerequation [1C–4C]:

$\begin{matrix}{{\gamma\; V} = {{\sum\limits_{n = 0}^{\infty}{T\left( {{- G_{0k}^{+}}T} \right)}^{n}} = {\sum\limits_{j = 1}^{\infty}{\lambda^{j}V_{j}}}}} & (254) \\{T = {\sum\limits_{n = 0}^{\infty}{\left( {\gamma\;{VG}_{0k}^{+}} \right)^{n}\gamma\; V}}} & (255)\end{matrix}$and this leads to

$\begin{matrix}{{\lambda\; T} = {\sum\limits_{n = 0}^{\infty}{\left( {\sum\limits_{j = 1}^{\infty}{\lambda^{j}V_{j}G_{0k}^{+}}} \right)^{n}{\sum\limits_{j^{\prime} = 1}^{\infty}{\lambda^{j^{\prime}}V_{j^{\prime}}}}}}} & (256)\end{matrix}$implying thenλ¹: T=V₁  (257)

$\begin{matrix}{{\lambda^{2}\text{:}0} = {V_{2} + {V_{1}G_{0k}^{+}V_{1}}}} & (258) \\{{\lambda^{3}\text{:}0} = {V_{3} + {V_{2}G_{0k}^{+}V_{1}G_{0k}^{+}V_{2}} + {V_{1}G_{0k}^{+}V_{1}G_{0k}^{+}V_{1}}}} & (259)\end{matrix}$etc. Again, one can show that in general,

$\begin{matrix}{V_{j} = {{- V_{j - 1}}G_{0k}^{+}V_{1}}} & (260)\end{matrix}$

However, it is crucial to recognize that{tilde over (V)}_(j)≠V_(j)  (261)because they correspond to orders of completely different parameters({tilde over (V)}_(j) is jth order in {tilde over (T)} while V_(j) isjth order in T). By Equation (245) above, it is clear that each separatefactor of T involves all orders of {tilde over (T)} and vice versa:[1−ikπ{tilde over (T)}|k><k|]T={tilde over (T)}  (262)so thatT=[1−ikπ{tilde over (T)}|k<>k|] ⁻¹ {tilde over (T)}  (263)

-   -    

$\begin{matrix}{= {\sum\limits_{n = 1}^{\infty}\;{\left( {{{- {ik}}\;\pi\;\overset{\sim}{T}}❘k} \right\rangle\left\langle {k❘} \right)^{n}\overset{\sim}{T}}}} & (264)\end{matrix}$Thus it is clear that {tilde over (V)}_(j) and V_(j) cannot be the same.

Now we ask how can one combine measured data with the Volterra inverseseries? We compute the back-scattering matrix element of Equation (250):<−k|{tilde over (T)}|k>≡{tilde over (T)}(−k,k)=<−k|{tilde over (V)}₁|k>≡{tilde over (V)} ₁(−k,k)  (265)But {tilde over (T)}(−k,k) is not directly measured. The far-fieldquantities typically measured are r(k)=−ikp(−k,k), the reflectionamplitude, and t(k)=1−ikπT(k,k), the transmission amplitude. By Equation(245), we writeT(−k,k)={tilde over (T)}( −k,k)−ikπ{tilde over (T)}(−k,k)T(k,k)  (266)so

$\begin{matrix}{{{\overset{\sim}{V}}_{1}\left( {{- k},k} \right)} = \frac{{ir}(k)}{k\;\pi\;{t(k)}}} & (267)\end{matrix}$This expression is inverse-Fourier transformed to the space-domain,yielding {tilde over (V)}₁(z). The result is

$\begin{matrix}{{{\overset{\sim}{V}}_{1}(z)} = {{\frac{2i}{\pi}{\int_{- \infty}^{+ \infty}\ {{\mathbb{d}k}\frac{{\mathbb{e}}^{{- 2}{ikz}}{r(k)}}{{kt}(k)}}}} = {{- \frac{4}{\pi}}{\int_{0}^{+ \infty}\mspace{7mu}{{\mathbb{d}k}\frac{1}{k}{Im}\frac{{\mathbb{e}}^{{- 2}{ikz}}{r(k)}}{t(k)}}}}}} & (268)\end{matrix}$One obtains the higher order {tilde over (V)}_(j)(z) according to

$\begin{matrix}{{{\overset{\sim}{V}}_{j}(z)} = {\int_{- \infty}^{+ \infty}\mspace{7mu}{{\mathbb{d}\left( {2k} \right)}{\mathbb{e}}^{{- 2}{\mathbb{i}}\;{kz}}\left\langle {{- k}❘{{{\overset{\sim}{V}}_{j - 1}{\overset{\sim}{G}}_{0k}{\overset{\sim}{V}}_{1}}❘k}} \right\rangle}}} & (269) \\{\mspace{56mu}{= {\frac{1}{2\pi}{\int_{- \infty}^{+ \infty}\mspace{7mu}{{\mathbb{d}\left( {2k} \right)}{\mathbb{e}}^{{- 2}{\mathbb{i}}\;{kz}}{\int_{- \infty}^{+ \infty}\mspace{7mu}{{\mathbb{d}z^{\prime}}{\int_{- \infty}^{+ \infty}\mspace{7mu}{{\mathbb{d}z^{''}}{\mathbb{e}}^{{\mathbb{i}}\;{{kz}{({z^{\prime} + z^{''}})}}}{{\overset{\sim}{V}}_{j - 1}\left( z^{\prime} \right)}{{\overset{\sim}{G}}_{0k}\left( {z^{\prime},z^{''}} \right)}{{\overset{\sim}{V}}_{1}\left( z^{''} \right)}}}}}}}}}} & (270)\end{matrix}$Again, it is instructive to carry out the application to scattering bythe Dirac delta function interaction discussed earlier. In that case, wehave

$\begin{matrix}{{r(k)} = {\frac{{- {ik}}\;\gamma}{2 + {{ik}\;\gamma}}{\mathbb{e}}^{2{\mathbb{i}}\;{kz}_{0}}}} & (271) \\{{t(k)} = \frac{2}{\left( {2 + {{ik}\;\gamma}} \right)}} & (272)\end{matrix}$It can then be shown that{tilde over (V)} ₁(z)=γδ(z−z ₀)  (273)

The second order corrections is given by

$\begin{matrix}{{{\overset{\sim}{V}}_{2}(z)} = {\gamma^{2}{\int_{- \infty}^{+ \infty}\mspace{7mu}{{\mathbb{d}\left( {2k} \right)}{\mathbb{e}}^{{- 2}{\mathbb{i}}\;{kz}}{\int_{- \infty}^{+ \infty}{{\mathbb{d}z^{\prime}}{\int_{- \infty}^{+ \infty}{{\mathbb{d}z^{''}}{\mathbb{e}}^{{\mathbb{i}}\;{{kz}{({z^{\prime} + z^{''}})}}}{\delta\left( {z^{\prime} - z_{0}} \right)}{{\overset{\sim}{G}}_{0k}\left( {z^{\prime},z^{''}} \right)}{\delta\left( {z^{''} - z_{0}} \right)}}}}}}}}} & (274) \\{\mspace{59mu}{= {{\gamma^{2}{\int_{- \infty}^{+ \infty}{{\mathbb{d}\left( {2k} \right)}{\mathbb{e}}^{{- 2}{\mathbb{i}}\;{ikz}}{{\overset{\sim}{G}}_{0k}\left( {z_{0},z_{0}} \right)}}}} \equiv 0}}} & (275)\end{matrix}$It should be clear that all {tilde over (V)}_(j) vanish for j≧2. Weconclude that the Volterra inverse scattering series converges to theexact answer in a single term, in the same manner as the forwardVolterra series for the Dirac delta function interaction. We note that acrucial change from the Born-Neumann approach to theLippmann-Schwinger-based inversion is that now, we require both r(k) andt(k) to use the Volterra inverse series!

Before leaving this example, we point out that Razavy [4C] hasconsidered the interaction 2λδ(x) within the Lippmann-Schwinger-basedinverse scattering series. Again, analytical results are obtained andthe exact result is equal to the sum of the V₁ and V₂ terms. However,the higher order terms were not evaluated, they do not vanish, andinvolve alternating signs. Thus, it appears that the series is onlyconditionally convergent, depending on the order in which the terms aregrouped and summed.

E. The Volterra Series for Non-Local Potentials

Up to this point, we have established that the Volterra property isshared by the coordinate representation matrix element, <z|{tilde over(T)}|z′>, if the interaction is local. In fact, we now show that thisresult is true for all non-local interactions so long as they alsopossess the Volterra kernel property. We recall that the exact solutionfor T is{tilde over (T)}=γV+γ ² V{tilde over (G)}V  (276)where{tilde over (G)}={tilde over (G)} ₀ +{tilde over (G)} ₀ γV{tilde over(G)}  (277)and{tilde over (G)}( z,z′)=0, z≧z′  (278)Now we do not restrict V to be local but we require thatV(z,z′)=0, z>z′  (279)

We want to prove that it remains true that {tilde over (T)}(z,z′)=0,z>z′. Our equation now is

$\begin{matrix}{{\overset{\sim}{T}\left( {z,z^{\prime}} \right)} = {{\gamma\;{V\left( {z,z^{\prime}} \right)}} + {\gamma^{2}{\int_{z}^{+ \infty}{{\mathbb{d}z^{''}}{\int_{- \infty}^{z^{\prime}}{{\mathbb{d}z^{\prime\prime\prime}}{V\left( {z,z^{''}} \right)}{\overset{\sim}{G}\left( {z^{''},z^{\prime\prime\prime}} \right)}{V\left( {z^{\prime\prime\prime},z^{\prime}} \right)}}}}}}}} & (280)\end{matrix}$where we use the facts that z″ must be greater than z and z′ must begreater than z′″, due to the presence of the factors V(z, z″) and V(z′″,z′). Now suppose that z>z′. The first term on the RHS of Equation (280)vanishes for this condition. But the second term only has non zerocontributions for z′>z′″>z″>z since all terms involving z″>z′″ vanishdue to the factor {tilde over (G)}(z″, z′″). Therefore, the only nonzeroterms contradict the condition z>z′. We conclude that {tilde over(T)}(z, z′)=0 if z>z′, if the non-local potential has the same property.

The analysis that the kernel of the inverse scattering equation has theVolterra property and, therefore, a Fredholm determinant that equals onecan be carried out and we do not write it explicitly here. If one hasthe most general form of non-local potential, V(z, z′), for 1Dscattering, then it turns out that the inversion requires measuring boththe far and near fields. Clearly, the operator equations obtained fromthe inversion of k²γV in terms of the {tilde over (V)}_(j)'s holdregardless of whether the potential is local or non-local. This is alsotrue of the absolute convergence of the series for k²γV in terms of the{tilde over (V)}_(j) (provided V itself has a Volterra-kernelstructure). However, if the non-local potential has the general formV(z,z′), then Equation (250) becomes{tilde over (T)}(k′,k)={tilde over (V)}(k′,k)  (281)with k′ and k independent of one another. Then one uses Equation (245)to determine the off-shell elements of {tilde over (T)} in terms of thephysical T-matrix elements:T(k′,k)={tilde over (T)}(k′,k)−ikπ{tilde over (T)}(k′,k)T(k,k)  (282)so that

$\begin{matrix}{{\overset{\sim}{T}\left( {k^{\prime},k} \right)} = \frac{T\left( {k^{\prime},k} \right)}{1 - {{ik}\;\pi\;{T\left( {k,k} \right)}}}} & (283)\end{matrix}$Thus, knowledge of the on- and half-off-shell T-matrix elements enablesone to determine the corresponding elements of the {tilde over(T)}-matrix. Then inverse Fourier transforming on both k′ and kindependently yields {tilde over (V)}₁(z,z′), which enables one todetermine all higher {tilde over (V)}_(j)(z,z′), j>1.

It should be clear that scattering interactions that can be expressed inthe form

${V(z)}\frac{d^{n}}{d\; z^{n}}{\delta\left( {z - z^{\prime}} \right)}$will also produce Volterra kernels (that is, interactions involvingderivatives of the field). Thus, the range of systems for which ourresults hold is very broad.XV. Application to the Square Well or Barrier

As a second example, we present the results of the Volterra-basedinverse series for acoustic scattering by a finite width well orbarrier. Again the reflection and transmission amplitudes can beobtained analytically, as can the various {tilde over (V)}_(j) terms inthe power series for the potential. One can show that

$\begin{matrix}{{r(k)} = \frac{V_{0}}{\left( {2 - V_{0}} \right) + {2i\sqrt{1 - V_{0}}{\cot\left( {{ak}\sqrt{1 - V_{0}}} \right)}}}} & (284) \\{{t(k)} = {\frac{2\sqrt{1 - V_{0}}i\;{\mathbb{e}}^{{- {\mathbb{i}}}\;{ka}}}{V_{0}{\sin\left( {{ak}\sqrt{1 - V_{0}}} \right)}}{r(k)}}} & (285)\end{matrix}$In the case of V₀<0, one can have any finite value for the magnitude ofthe interaction (corresponding to any finite increase in the velocity ofsound in the medium). In the case of a barrier, 0<V₀<1; otherwise oneencounters an infinite (V₀=1) or pure imaginary (V₀>1) velocity ofsound. It follows that

$\begin{matrix}{{{\overset{\sim}{V}}_{1}\left( {2k} \right)} = {\frac{V_{0}}{2\pi\; k\sqrt{1 - V_{0}}}{\sin\left( {{ak}\sqrt{1 - V_{0}}} \right)}{\mathbb{e}}^{{\mathbb{i}}\;{ka}}}} & (286)\end{matrix}$The {tilde over (V)}₁(z) is then

$\begin{matrix}{{{\overset{\sim}{V}}_{1}(z)} = {\frac{V_{0}}{\pi\sqrt{1 - V_{0}}}{\int_{- \infty}^{+ \infty}\ {{\mathbb{d}k}\frac{\sin\left( {{ak}\sqrt{1 - V_{0}}} \right)}{k}{\mathbb{e}}^{{\mathbb{i}}\; k\;{({a - {2z}})}}}}}} & (287)\end{matrix}$which is recognized as the Fourier transform of the sinc-function. Thisis well known to be a square well or barrier:

-   -    

$\begin{matrix}{{{\overset{\sim}{V}}_{1} = \frac{V_{0}}{\sqrt{1 - V_{0}}}},{{{a - {2z}}} < {a\sqrt{1 - V_{0}}}}} & (288) \\{\mspace{31mu}{{= 0},{{all}\mspace{20mu}{other}\mspace{20mu} z}}} & (289)\end{matrix}$

Rearranging, we find that the region where {tilde over (V)}₁ is nonzeroisz_(min)<z<z_(max)  (290)

-   -    

$\begin{matrix}{z_{\min} = {\frac{a}{2}\left( {1 - \sqrt{1 - V_{0}}} \right)}} & (291) \\{z_{\max} = {\frac{a}{2}\left( {1 + \sqrt{1 - V_{0}}} \right)}} & (292)\end{matrix}$

For a barrier, 0<V₀<1 and the first order result has a higher barrierthan the true one. For a well, V₀<0 and the first order result isshallower than the true one. Thus, although the first order result hasthe correct analytical form of a square well or barrier, it has incorrect width and height (or depth). However, the explicit form of theresult is such that it is trivial to obtain the exact potential fromz_(min) and z_(max). It can be seen that

$\begin{matrix}{V_{0} = {1 - \left( \frac{z_{\max} - z_{\min}}{z_{\max} + z_{\min}} \right)^{2}}} & (293)\end{matrix}$anda=z _(max) +z _(min)  (294)

$\begin{matrix}{a = {\frac{2z_{\min}}{1 - \sqrt{1 - V_{0}}} = \frac{2z_{\max}}{1 + \sqrt{1 - V_{0}}}}} & (295)\end{matrix}$

These exact, analytical expressions are found to work very well incomputational studies as well. Thus, it is not necessary to evaluate the{tilde over (V)}_(j) beyond j=1 in order to obtain the exact parametersfor a square well or barrier interaction. Even so, these higher orderterms an also be evaluated analytically.

These results can again be compared to those obtained using theFredholm-based Born-Neumann inverse scattering series. Razavy [4C] hasalso obtained an expression for the V₁ term. In fact, the result is ofthe form of an infinite series, so a closed expression has not beenpossible. This also prevented him from obtaining higher ordercorrections. However, the structure manifested at the first order is nota simple square well but rather an infinite sequence of steps ofdecreasing magnitude. Razavy does not consider the convergence of theseries. Despite these qualitatively incorrect features, it isnever-the-less possible to use Razavy's result to determine the squareinteraction parameters exactly. This is because the terms in theinfinite series permit one to obtain the correct V₀ and α-parameter fromthe first of the infinite series of steps. However, because theFredholm-based inversion produces unphysical artifacts that are absentfrom the Volterra-based results, the latter provides a more robustframework for an inversion when one has an interaction that does notyield an explicit formula for the various terms in the series.

It is remarkable that the Volterra-based inverse scattering series forboth of these simple potentials is able to provide either the exactanswer or the exact functional form of the interaction with only thefirst order term. Furthermore, the fat that all higher order terms canbe evaluated analytically is very useful. We stress that these resultsare consequences of the fact that the Volterra-based inversion makes useof both the reflection and transmission information.

XVI. Conclusions and Future Work

In this portion of the application, we have used the fat that theacoustic scattering Lippmann-Schwinger integral equation (in 1D)involving the causal (or anti-causal) Green's function can berenomialized to write it as a Volterra integral equation. Such equationspossess the best possible convergence behavior under Born-Neumanniteration. Furthermore, for a wide class of interactions (local,differential, or non-local but with the Volterra property), theauxiliary transition operator also possesses the Volterra property.Consequently, the inverse acoustic scattering series obtained byreverting the Volterra-based series in terms of {tilde over (V)}_(j)also converges absolutely and uniformly for all |γ|<∞. This does not, ofcourse, ensure that the rate of convergence is conveniently rapid. It iswell-known that an absolutely convergent series can be rearranged orgrouped in any manner without affecting its convergence [Kaplan [16C]].Of course, this is not true for divergent or conditionally convergentseries. In the case of seismic scattering, one may expect the changes inthe velocity of sound to be modeled reasonably by piece-wise constantinteractions sine the distance over which there can be large changesshould be small compared to distances over which the sonic speed changesless rapidly.

Our results show that a Volterra-based inversion can be done as a singlecomprehensive task, provided one has both the reflection andtransmission amplitudes as functions of k. Indeed, all 1D scatteringproblems that can be formulated in a Lippmann-Schwinger framework havenow been shown to be invertible, given the r(k) and t(k). In subsequentwork we shall consider this approach for scattering in higher dimensionsas well. The implications for various applications such as medialimaging, seismic exploration, non-destructive testing, etc. areundercurrent study and results will be reported as they are obtained.

By appropriate use of Equation (194), we have been able to express theVolterra-based inversion in a form that requires only r(k) as input,rather than both r(k) and t(k). This is an important reduction in theexperimental data required to apply our approach. It has been pointedout to us that for the Dirac delta interaction, an approach based on theHeitler damping relation also yields the exact result [17C]. Theapproach is for evaluating a first order approximation only. A completediscussion of the relation to the present approach will be givenelsewhere.

Appendix C

In this Appendix we give a few more details regarding the Fredholmsolution of Equation (28),

$\begin{matrix}{{p_{k}(z)} = {{\mathbb{e}}^{{\mathbb{i}}\;{kz}} + {\int_{z}^{+ \infty}\mspace{7mu}{{\mathbb{d}z^{''}}k^{2}\gamma\;{K\left( {z,z^{''}} \right)}{p_{k}\left( z^{''} \right)}}}}} & (296)\end{matrix}$where K(z, z″) is defined in Equations (201)–(202). The solution may bewritten as

$\begin{matrix}{p_{k} = {{\mathbb{e}}^{{\mathbb{i}}\;{kz}} + {\int_{0}^{+ \infty}{{\mathbb{d}z^{\prime}}\frac{D\left( {z,z^{\prime}} \right)}{D}{\mathbb{e}}^{{\mathbb{i}}\;{kz}^{\prime}}}}}} & (297)\end{matrix}$where

$\begin{matrix}\begin{matrix}{D = {1 - {k^{2}\gamma{\int_{0}^{+ \infty}\mspace{7mu}{{\mathbb{d}z}\; K\left( {z,z} \right)}}} +}} \\{{\frac{\left( {k^{2}\gamma} \right)^{2}}{2!}{\int_{0}^{+ \infty}{{\mathbb{d}z}{\int_{0}^{+ \infty}{{\mathbb{d}z^{\prime}}{\det\begin{pmatrix}{K\left( {z,z} \right)} & {K\left( {z,z^{\prime}} \right)} \\{K\left( {z^{\prime},z} \right)} & {K\left( {z^{\prime},z^{\prime}} \right)}\end{pmatrix}}}}}}} - \cdots}\end{matrix} & (298) \\{and} & \; \\\begin{matrix}{{D\left( {z,z^{\prime}} \right)} = {{k^{2}\gamma\;{K\left( {z,z^{\prime}} \right)}} -}} \\{{\left( {k^{2}\gamma} \right)^{2}{\int_{0}^{+ \infty}{{\mathbb{d}z^{''}}{\det\begin{pmatrix}{K\left( {z,z^{\prime}} \right)} & {K\left( {z,z^{''}} \right)} \\{K\left( {z^{''},z^{\prime}} \right)} & {K\left( {z^{''},z^{''}} \right)}\end{pmatrix}}}}} + \cdots}\end{matrix} & (299)\end{matrix}$

Note that K(z,z) vanishes so long as V(z) is not too singular.Therefore, all diagonal terms in the determinants appearing in Equation(298) for D vanish. All other terms vanish, as discussed in the textabove, sine they are of the form Tr(K″). Consequently, D=1. Now considerthe integral

∫₀^(∞) 𝕕z^(′)D(z, z^(′))𝕖^(𝕚 kz^(′)).We assume, for simplicity and convenience, that the potential hascompact support on the domain [0, Z], and that it is bounded. For anyvalue of k and y, we conclude that k²γK(z, z′)<|a|, where a is somefinite number. By Hadamard's theorem [6, 14], the value of an nth orderdeterminant formed from such elements is bounded by |a|^(n)n^(n/2). Thenthe nth term, say t_(n), in Equation (299) is bounded by

$\begin{matrix}{t_{n} < {\frac{1}{n!}Z^{n}{a}^{n}n^{n/2}}} & (300)\end{matrix}$

Using Stirling's approximation, one has that

$\begin{matrix}{t_{n} < \frac{Z^{n}{a}^{n}}{{\mathbb{e}}^{- n}n^{n/2}n^{1/2}}} & (301)\end{matrix}$By the root test (Kaplan [16]), we see that

$\begin{matrix}{{\lim\limits_{n->\infty}\left( t_{n} \right)^{1/n}} = {{\lim\limits_{n->\infty}\left( \frac{Z{a}{\mathbb{e}}}{n^{1/2}n^{{1/2}n}} \right)} = 0}} & (302)\end{matrix}$The radius of convergence is one divided by this limit so we concludethat the series for D (z, z′) converges absolutely independent of thestrength of the coupling parameter, γ or the value of k.

The most robust treatment of the inverse acoustic scattering problem isthat based on the reversion of the Born-Neumann series solution of theLippmann-Schwinger equation. An important issue for this approach toinversion is the radius of convergence of the Born-Neumann series forFredholm integral kernels, and especially for acoustic scattering forwhich the interaction depends on the square of the frequency. Bycontrast, it is well known that the Born-Neumann series for the Volterraintegral equations in quantum scattering are absolutely convergent,in-dependent of the strength of the coupling characterizing theinteraction. The transformation of the Lippmann-Schwinger equation froma Fredholm to a Volterra structure by renormalization has beenconsidered previously for quantum scattering calculations andelectromagnetic scattering. In this portion of the application, weemploy the renormalization technique to obtain a Volterra equationframework for the inverse acoustic scattering series, proving that thisseries also converges absolutely in the entire complex plane of couplingconstant and frequency values. The present results are for acousticscattering in one dimension but the method is general. The approach isillustrated by applications to two simple one dimensional models foracoustic scattering.

REFERENCES

-   [1C] A. B. Weglein, K. H. Matson, D. J. Foster, P. M. Carvalho, D.    Corrigan, S. A. Shaw, Imaging and inversion at depth without a    velocity model: theory, concepts and initial evaluation, So.    Exploration Geophysics 2000, Expanded Abstracts, Calgary,    Calif.; A. B. Weglein and R. H. Stolt, Migration-inversion    revisited, in The Leading Edge (1999) 950. See also the subseries    approach to removing multiples from seismic data in A. B.    Weglein, F. A. Gasparotto, P. M. Carvalho, and R. H. Stolt,    Geophysics 62, 1775 (1997).-   [2C] R. Jost and W. Kohn, Phys. Rev. 87, 977 (1952).-   [3C] H. E. Moses, Phys. Rev. 102, 559 (1956).-   [4C] M. Razavy, J. Acoust. So. Am. 58, 956 (1975).-   [5C] P. M. Morse and H. Feshbah, Methods of Theoretical Physics    (McGraw-Hill, New York, 1953).-   [6C] R. G. Newton, Scattering Theory of Waves and Particles    (Springer-Verlag, New York, 1982).-   [7C] M. L. Goldberger and K. M. Watson, Collision Theory (Wiley, New    York, 1964).-   [8C] R. T. Prosser, J. Math. Phys. 10, 1819 (1969); ibid., 17, 1775    (1976); ibid. 21, 2648 (1980).-   [9C] R. Courant and D. Hilbert, Methods of Mathematical Physics    (Interscience, New York, 1953).-   [10C] W. N. Sams and D. J. Kouri, J. Chem. Phys. 51, 4809 and 4815    (1969).-   [11C] D. J. Kouri, J. Math. Phys. 14, 1116 (1973).-   [12C] To be published.-   [13C] A. B. Weglein, private communication.-   [14C] L. S. Rodberg and R. M. Thaler, The Quantum Theory of    Scattering (Academic Press, New York, 1967).-   [15C] J. Mathews and R. L. Walker, Mathematical Methods of Physics    (Benjamin, New York, 1965).-   [16C] W. Kaplan, Advanced Calculus (Addison-Wesley, Reading, Mass.,    1952).-   [17C] A. J. Devaney and A. B. Weglein, J. Inverse Problems 5, 49    (1989).

3D Acoustic Spherical Interaction and Non-Spherical Interaction

The following equation is the general acoustic case:

$\begin{matrix}{{kr}\left\lbrack {{P_{lik}^{+}(r)} = {{j_{1}({kr})} - {\frac{2{mk}}{\hslash^{2}}{\int_{0}^{\infty}{{\mathbb{d}r^{\prime}}r^{\prime 2}{h_{l}^{+}\left( {kr}_{>} \right)}{j_{l}\left( {kr}_{<} \right)}k^{2}{V\left( r^{\prime} \right)}{P_{lik}^{+}\left( r^{\prime} \right)}}}}}} \right\rbrack} & (303)\end{matrix}$Introduction—Ricetti Functions

Let

H_(l)⁺ = krh_(l)⁺and J_(l)=krj_(l) for m=½ and

=1 for the acoustic case.

$\begin{matrix}{{P_{lik}^{+}(r)} = {{krP}_{lk}^{+} = {{J_{l}({kr})} - {k{\int_{0}^{\infty}{{\mathbb{d}r^{\prime}}{H_{l}^{+}\left( {kr}_{>} \right)}{J_{l}\left( {kr}_{<} \right)}{V\left( r^{\prime} \right)}{P_{lik}^{+}\left( r^{\prime} \right)}}}}}}} & (304) \\{{P_{lik}^{+}(r)} = {{{J_{l}({kr})}\left\lbrack {1 - {k{\int_{0}^{\infty}{{\mathbb{d}r^{\prime}}{H_{l}^{+}({kr})}{V\left( r^{\prime} \right)}{P_{lik}^{\prime}\left( r^{\prime} \right)}}}}} \right\rbrack} - {k{\int_{0}^{r}{{\mathbb{d}{r^{\prime}\left\lbrack {{{N_{l}({kr})}{J_{l}\left( {kr}^{\prime} \right)}} - {{J_{l}({kr})}{N_{l}\left( {kr}^{\prime} \right)}}} \right\rbrack}}{V\left( r^{\prime} \right)}{P_{lik}^{+}\left( r^{\prime} \right)}}}}}} & (305)\end{matrix}$where H_(l) ⁺=N_(l)+iJ_(l). Then T_(l)|J_(lk)>=VP_(lk) ⁺(r)

$\begin{matrix}{T_{l} = {V + {{VG}_{10k}^{+}T_{l}}}} & (306) \\{G_{10k}^{+} = {{- {{kH}_{l}^{+}\left( {kr}_{>} \right)}}{J_{l}\left( {kr}_{<} \right)}}} & (307) \\{\left. {G_{10k}^{+} = {{{\overset{\sim}{G}}_{10k} - k}❘J_{lk}}} \right\rangle\left\langle {H_{lk}^{+}❘} \right.} & (308) \\{T_{l} = {{{V\left\lbrack {{1 - k}❘J_{lk}} \right\rangle}\left\langle {H_{lk}^{+}❘T_{l}} \right\rbrack} + {V{\overset{\sim}{G}}_{10k}T_{l}}}} & (309)\end{matrix}$so that we can derive the following

$\begin{matrix}{T_{l} = {{T_{l}\left\lbrack {{1 - k}❘J_{lk}} \right\rangle}\left\langle {H_{lk}^{+}❘T_{l}} \right\rbrack}} & (310)\end{matrix}${tilde over (T)} ₁ =V+V{tilde over (G)} _(l0k) {tilde over (T)}_(l)  (311)

$\begin{matrix}{{V = {\sum\limits_{j}^{\;}\;{\overset{\sim}{V}}_{j}}},{{\overset{\sim}{V}}_{1} = {\overset{\sim}{T}}_{l}},{{\overset{\sim}{V}}_{j} = {{\overset{\sim}{V}}_{j - 1}{\overset{\sim}{G}}_{l0k}{\overset{\sim}{V}}_{1}}},{j \geq 2}} & (312) \\{{P_{\overset{\rightarrow}{k}}^{+}\left( \overset{\rightarrow}{r} \right)} = {{4\pi{\sum\limits_{l\;\mu}^{\;}\;{i^{l}{Y_{l\;\mu}^{*}\left( \hat{k} \right)}{Y_{l\;\mu}\left( \hat{r} \right)}{P_{lk}^{+}(r)}}}}\mspace{59mu} = {\frac{4\pi}{kr}{\sum\limits_{l\;\mu}^{\;}\;{i^{l}{Y_{l\;\mu}^{*}\left( \hat{k} \right)}{Y_{l\;\mu}\left( \hat{r} \right)}{P_{lk}^{+}(r)}}}}}} & (313) \\{\left\lbrack {P_{\overset{\rightarrow}{k}}^{+} - {\mathbb{e}}^{{\mathbb{i}}{\overset{\rightarrow}{k} \cdot \overset{\rightarrow}{r}}}} \right\rbrack = {P_{\overset{\rightarrow}{k},{sw}}^{+}\left( \overset{\rightarrow}{r} \right)}} & (314)\end{matrix}$where

$P_{\overset{\rightarrow}{k},{sw}}^{+}\left( \overset{\rightarrow}{r} \right)$is data from spectra.

At r outside the range of V(r), we have:

$\begin{matrix}{{P_{\overset{\rightarrow}{k},{sw}}^{+}\left( \overset{\rightarrow}{r} \right)} = {{- \frac{4\pi}{r}}{\sum\limits_{l\;\mu}^{\;}\;{i^{l}{H_{l}^{+}({kr})}{Y_{l\;\mu}^{*}\left( \hat{k} \right)}{Y_{l\;\mu}\left( \hat{r} \right)}{\int_{0}^{R}\mspace{7mu}{{\mathbb{d}r^{\prime}}{J_{l}\left( {kr}^{\prime} \right)}{V\left( {kr}^{\prime} \right)}{P_{lk}^{+}\left( r^{\prime} \right)}}}}}}} & (315)\end{matrix}$

We now define:

$\begin{matrix}{T_{l}^{(1)} = {{\int_{0}^{R}\mspace{7mu}{{\mathbb{d}r^{\prime}}{J_{l}\left( {kr}^{\prime} \right)}{V\left( r^{\prime} \right)}{P_{lk}^{+}\left( r^{\prime} \right)}}} = \left\langle {J_{lk}❘{T_{l}❘J_{lk}}} \right\rangle}} & (316) \\{{Data} = {{- \frac{4\pi}{r}}{\sum\limits_{l}^{\;}\;{\left( {{2l} + 1} \right){P_{l}\left( {\overset{\rightarrow}{k} \cdot \overset{\rightarrow}{r}} \right)}{H_{l}^{+}({kr})}T_{l}^{(1)}}}}} & (317) \\{T_{l}^{(1)} = {{\overset{\sim}{T}}_{l}^{(1)}\left\lbrack {1 - {kT}_{l}^{(2)}} \right\rbrack}} & (318) \\{where} & \; \\{T_{l}^{(2)} \equiv \left\langle {H_{lk}^{+}{T_{l}}J_{lk}} \right\rangle} & (319) \\{and} & \; \\{T_{l}^{(2)} = {{\overset{\sim}{T}}_{l}^{(2)}\left\lbrack {1 - {kT}_{l}^{(2)}} \right\rbrack}} & (320) \\{T_{l}^{(2)} = \frac{{\overset{\sim}{T}}_{l}^{(2)}}{1 + {k{\overset{\sim}{T}}_{l}^{(2)}}}} & (321)\end{matrix}$which yields

$\begin{matrix}{T_{l}^{(1)} = {{{\overset{\sim}{T}}_{l}^{(1)}\left\lbrack {1 - \frac{k{\overset{\sim}{T}}_{l}^{(2)}}{1 + {k{\overset{\sim}{T}}_{l}^{(2)}}}} \right\rbrack} = \frac{{\overset{\sim}{T}}_{l}^{(1)}}{1 + {k{\overset{\sim}{T}}_{l}^{(2)}}}}} & (322) \\{{Data} = {{- \frac{4\pi}{r}}{\sum\limits_{l}^{\;}\;{\left( {{2l} + 1} \right){P_{l}\left( {\overset{\rightarrow}{k} \cdot \overset{\rightarrow}{r}} \right)}{H_{l}^{+}({kr})}\frac{{\overset{\sim}{T}}_{l}^{(2)}}{\left( {1 + {k{\overset{\sim}{T}}_{l}^{(2)}}} \right)}}}}} & (323)\end{matrix}$

Then to first order, {tilde over (V)}₁={tilde over (T)}_(l), so that

$\begin{matrix}{{Data} = {{- \frac{4\pi}{r}}{\sum\limits_{l}^{\;}\;{\left( {{2l} + 1} \right){P_{l}\left( {\overset{\rightarrow}{k} \cdot \overset{\rightarrow}{r}} \right)}{H_{l}^{+}({kr})}\frac{{\overset{\sim}{V}}_{1l}^{(1)}}{\left( {1 + {k\;{\overset{\sim}{V}}_{1l}^{(2)}}} \right)}}}}} & (324)\end{matrix}$where

$\begin{matrix}{{\overset{\sim}{V}}_{1l}^{(1)} = \left\langle {J_{lk}❘{{\overset{\sim}{V}}_{1}❘J_{lk}}} \right\rangle} & (325) \\{{\overset{\sim}{V}}_{1l}^{(2)} \equiv \left\langle {H_{lk}^{+}{{\overset{\sim}{V}}_{1}}J_{lk}} \right\rangle} & (326)\end{matrix}$

The higher order corrections are given by:

$\begin{matrix}\begin{matrix}{{{\overset{\sim}{V}}_{2}(r)} = \left( {{- {\overset{\sim}{V}}_{1}}{\overset{\sim}{G}}_{l0k}{\overset{\sim}{V}}_{1}} \right)_{r}} \\\vdots \\{{{\overset{\sim}{V}}_{j}(r)} = \left( {{- {\overset{\sim}{V}}_{j - 1}}{\overset{\sim}{G}}_{l0k}{\overset{\sim}{V}}_{1}} \right)_{r}}\end{matrix} & \left( {327a\mspace{11mu}\ldots} \right)\end{matrix}$The higher order corrections converge absolutely and uniformly allowinganalysis of inverse scattering spectral data to any level of accuracy.Additional Notation for the Non-Spherical Case

$\begin{matrix}\begin{matrix}{{{kr}\left\lbrack {P^{+}\left( {{l\;\mu}❘{{l^{\prime}\mu^{\prime}}❘r}} \right)} \right\rbrack} = {{\delta_{{ll}^{\prime}}\delta_{{\mu\mu}^{\prime}}{j_{l}({kr})}} -}} \\{\frac{2{mk}}{\hslash^{2}}{\sum\limits_{l^{''}\mu^{''}}^{\;}\;{\int{{\mathbb{d}{\overset{\rightarrow}{r}}^{\prime}}{Y_{l\mu}^{*}\left( {\hat{r}}^{\prime} \right)}{h_{l}^{+}\left( {kr}_{>} \right)}}}}} \\{{j_{l}\left( {kr}_{<} \right)}k^{2}{V\left( {\hat{r}}^{\prime} \right)}{Y_{l^{''}\mu^{''}}\left( {\hat{r}}^{\prime} \right)}{P^{+}\left( {l^{''}\mu^{''}{{l^{\prime}\mu^{\prime}}}r^{\prime}} \right)}}\end{matrix} & (328)\end{matrix}$

$\begin{matrix}{\left( {\left. {\underset{\_}{J}}_{k} \right\rangle\left\langle {\underset{\_}{H}}_{k}^{+} \right.} \right)_{\;{l\;\mu\; l^{\prime}\mu^{\prime}}} = {\delta_{{ll}^{\prime}}\delta_{{\mu\mu}^{\prime}}\left. J_{lk} \right\rangle\left\langle H_{lk}^{+} \right.}} & (329)\end{matrix}$

$\begin{matrix}\begin{matrix}{{P_{k}^{\prime}\left( {l\;\mu{{l^{\prime}\mu^{\prime}}}r} \right)} = {{\delta_{{ll}^{\prime}}\delta_{{\mu\mu}^{\prime}}{J_{l}({kr})}} -}} \\{k{\sum\limits_{l^{''}\mu^{''}}{\int_{0}^{\infty}{{\mathbb{d}r^{\prime}}r^{\prime 2}{H_{l}^{+}\left( {kr}_{>} \right)}{J_{l}\left( {kr}_{<} \right)}}}}} \\{{V\left( {l\;\mu{{l^{''}\mu^{''}}}r^{\prime}} \right)}{P_{k}^{+}\left( {l^{''}\mu^{''}{{l^{\prime}\mu^{\prime}}}r^{\prime}} \right)}}\end{matrix} & (330)\end{matrix}$

$\begin{matrix}{\left. {{T\left( {l\;\mu\left. {l^{\prime}\mu^{\prime}} \right)} \right.}J_{l^{\prime}k}} \right\rangle = {\sum\limits_{l^{''}\mu^{''}}{{V\left( {l\;\mu\left. {l^{''}\mu^{''}} \right)} \right.}{P_{k}^{+}\left( {l^{''}\mu^{''}\left. {l^{\prime}\mu^{\prime}} \right)} \right\rangle}}}} & (331) \\{{{\underset{\_}{\underset{\_}{V}}\;{\underset{\_}{P}}_{k}^{+}} = {\underset{\_}{\underset{\_}{T}}{\underset{\_}{J}}_{k}}}{{{\underset{\_}{P}}_{k}^{+}\left( {l^{\prime}\mu^{\prime}} \right)} = {{{\underset{\_}{J}}_{k}\left( {l^{\prime}\mu^{\prime}} \right)} - {k\;{\underset{\_}{\underset{\_}{G}}}_{0k}^{+}\underset{\_}{\underset{\_}{V}}{{\underset{\_}{P}}_{k}^{+}\left( {l^{\prime}\mu^{\prime}} \right)}}}}\underset{\_}{\underset{\_}{T}} = {\underset{\_}{\underset{\_}{V}} + {\underset{\_}{\underset{\_}{V}}{\underset{\_}{\underset{\_}{G}}}_{0k}^{+}\underset{\underset{\_}{\_}}{T}}}} & \left( {332\text{a-c}} \right)\end{matrix}$diagonal matrix elements are given by:

$\begin{matrix}\begin{matrix}{\underset{\_}{\underset{\_}{T}} = {\underset{\_}{\underset{\_}{\overset{\sim}{T}}}\left\lbrack {\underset{\_}{\underset{\_}{1}} - {k{\sum\limits_{l\;\mu}{\left. {{\underset{\_}{J}}_{k}\left( {l\;\mu} \right)} \right\rangle\left\langle {{\underset{\_}{H}}_{k}^{+}\left( {l\;\mu} \right)} \right.\underset{\_}{\underset{\_}{T}}}}}} \right\rbrack}} \\{\overset{\sim}{\underset{\_}{\underset{\_}{T}}} = {\underset{\_}{\underset{\_}{V}} + {\underset{\_}{\underset{\_}{V}}{\overset{\sim}{\underset{\_}{\underset{\_}{G}}}}_{0k}\overset{\sim}{\underset{\_}{\underset{\_}{T}}}}}}\end{matrix} & \left( {333\text{a-b}} \right) \\{{\sum\limits_{l^{''}\mu^{''}}{{V\left( {l\;\mu{{l^{''}\mu^{''}}}r} \right)}{P_{k}^{+}\left( {l^{''}\mu^{''}{{l^{\prime}\mu^{\prime}}}r} \right)}}} = {\sum\limits_{l^{''}\mu^{''}}{T\left( {{l^{''}\mu^{''}\left. {l^{\prime}\mu^{\prime}} \right)\delta_{l^{''}l^{\prime}}\delta_{\mu^{''}\mu^{\prime}}J_{l^{\prime}k}\left\langle {{{\underset{\_}{J}}_{k}\left( {l\;\mu} \right)}{\underset{\_}{\underset{\_}{T}}}{{\underset{\_}{J}}_{k}\left( {l^{\prime}\mu^{\prime}} \right)}} \right\rangle} = {{\left\langle {{\underset{\_}{J}}_{k}{\overset{\sim}{\underset{\_}{\underset{\_}{T}}}}{\underset{\_}{J}}_{k}} \right\rangle\left\langle {{\underset{\_}{J}}_{k}\left( {l\;\mu} \right){\underset{\_}{\underset{\_}{T}}}{{\underset{\_}{J}}_{k}\left( {l^{\prime}\mu^{\prime}} \right)}} \right\rangle} = {T\left( {l\;\mu\left. {l^{\prime}\mu^{\prime}} \right)} \right.}}} \right.}}} & \left( {334\text{a-c}} \right)\end{matrix}$

We need to use the expression

$G_{0k}^{+} = {{\overset{\sim}{G}}_{0k} + O}$

$\begin{matrix}\begin{matrix}{{{- \frac{1}{4\pi}}\frac{{\mathbb{e}}^{{\mathbb{i}}\; k{{\overset{\rightarrow}{r} - {\overset{\rightarrow}{r}}^{\prime}}}}}{{\overset{\rightarrow}{r} - {\overset{\rightarrow}{r}}^{\prime}}}} = {{- k}{\sum\limits_{l\;\mu}{{Y_{l\;\mu}\left( \hat{r} \right)}{{Y_{l\;\mu}^{*}\left( {\hat{r}}^{\prime} \right)}^{*}\left\lbrack {{{n_{l}({kr})}{j_{l}\left( {kr}^{\prime} \right)}} -} \right.}}}}} \\{\left. {{j_{l}({kr})}{n_{l}\left( {kr}^{\prime} \right)}} \right\rbrack -} \\{k{\sum\limits_{l\;\mu}{{Y_{l\;\mu}\left( \hat{r} \right)}{Y_{l\;\mu}^{*}\left( {\hat{r}}^{\prime} \right)}^{*}{j_{l}({kr})}{h_{l}^{+}\left( {kr}^{\prime} \right)}}}}\end{matrix} & (335) \\{{{\overset{\sim}{G}}_{0k} = {{- k}{\sum\limits_{l\;\mu}\left\lbrack {{\left. {Y_{l\;\mu}n_{lk}} \right\rangle\left\langle {Y_{l\;\mu}j_{lk}} \right.} - {\left. {Y_{l\;\mu}j_{lk}} \right\rangle\left\langle {Y_{l\;\mu}n_{lk}} \right.}} \right\rbrack}}}{O = {{- k}{\sum\limits_{l\;\mu}{\left. {Y_{l\;\mu}j_{lk}} \right\rangle\left\langle {Y_{l\;\mu}h_{lk}^{+}} \right.}}}}} & \left( {336\text{a-b}} \right) \\{\begin{matrix}{T = {V + {{VG}_{0k}^{+}T}}} \\{= {V + {V\;{\overset{\sim}{G}}_{0k}T} + {VOT}}} \\{= {{V\left\lbrack {1 + {OT}} \right\}} + {V\;{\overset{\sim}{G}}_{0k}T}}}\end{matrix}{T = {\overset{\sim}{T}\left\lbrack {1 + {OT}} \right\rbrack}}{\overset{\sim}{T} = {V + {V\;{\overset{\sim}{G}}_{0k}\overset{\sim}{T}}}}} & \left( {337\text{a-c}} \right) \\\begin{matrix}{{T^{(1)}\left( {l\;\mu{{l^{\prime}\mu^{\prime}}}k} \right)} = {{{\overset{\sim}{T}}^{(1)}\left( {l\;\mu{{l^{\prime}\mu^{\prime}}}k} \right)} + \left\langle {Y_{l\;\mu}j_{lk}{{\overset{\sim}{T}{OT}}}Y_{l^{\prime}\mu^{\prime}}j_{l^{\prime}k}} \right\rangle}} \\{= {{{\overset{\sim}{T}}^{(1)}\left( {l\;\mu{{l^{\prime}\mu^{\prime}}}k} \right)} - {k{\sum\limits_{l^{''}\mu^{''}}{{\overset{\sim}{T}}^{(1)}\left( {l\;\mu{{l^{''}\mu^{''}}}k} \right)}}}}} \\{{\overset{\sim}{T}}^{(2)}\left( {l^{''}\mu^{''}{{l^{\prime}\mu^{\prime}}}k} \right)}\end{matrix} & (338) \\\begin{matrix}{{T^{(2)}\left( {l\;\mu{{l^{\prime}\mu^{\prime}}}k} \right)} = {{{\overset{\sim}{T}}^{(2)}\left( {l\;\mu{{l^{\prime}\mu^{\prime}}}k} \right)} -}} \\{k{\sum\limits_{l^{''}\mu^{''}}{{{\overset{\sim}{T}}^{(2)}\left( {l\;\mu{{l^{''}\mu^{''}}}k} \right)}{{\overset{\sim}{T}}^{(2)}\left( {l^{''}\mu^{''}{{l^{\prime}\mu^{\prime}}}k} \right)}}}}\end{matrix} & (339)\end{matrix}$Solving for T ⁽²⁾ and plugging into equation (339) for T⁽¹⁾(lμ|l′μ′|k).Then we take a first order treatment and higher order terms again resultfrom{tilde over (V)} _(j) =−{tilde over (V)} _(j−1) {tilde over (G)} _(0k){tilde over (V)} ₁  (340)where {tilde over (G)}_(0k) is as described previously leading to a dataconstruction shown below:

$\begin{matrix}{{f\left( {\overset{\rightarrow}{k} \cdot \overset{\rightarrow}{r}} \right)} = {\sum\limits_{l\;\mu}{\sum\limits_{l^{\prime}\mu^{\prime}}{{Y_{l\;\mu}^{*}\left( \hat{k} \right)}{Y_{l^{\prime}\mu^{\prime}}\left( \hat{r} \right)}{T\left( {l^{\prime}\mu^{\prime}{{l\;\mu}}k} \right)}}}}} & (341)\end{matrix}$Now we can solve the case to the first order where {tilde over(T)}={tilde over (V)}₁, implies

$\begin{matrix}\begin{matrix}{{{\overset{\sim}{T}}^{(2)}\left( {l\;\mu{{l^{\prime}\mu^{\prime}}}k} \right)} = \left\langle {Y_{l\;\mu}h_{lk}^{+}{{{\overset{\sim}{V}}_{1}\left( \overset{\rightarrow}{r} \right)}}Y_{l^{\prime}\mu^{\prime}}j_{lk}} \right\rangle} \\{= {\int{{\mathbb{d}\overset{\rightarrow}{r}}{Y_{l\;\mu}^{*}\left( \overset{\rightarrow}{r} \right)}{h_{l}^{+}({kr})}{{\overset{\sim}{V}}_{1}\left( \overset{\rightarrow}{r} \right)}{Y_{l^{\prime}\mu^{\prime}}\left( \overset{\rightarrow}{r} \right)}{j_{l}({kr})}}}}\end{matrix} & (342) \\\begin{matrix}{{T^{(2)}\left( {l\;\mu{{l^{\prime}\mu^{\prime}}}k} \right)} = {{{\overset{\sim}{V}}_{1}^{(2)}\left( {l\;\mu{{l^{\prime}\mu^{\prime}}}k} \right)} -}} \\{k{\sum\limits_{l^{''}\mu^{''}}{{{\overset{\sim}{V}}_{1}^{(2)}\left( {l\;\mu{{l^{''}\mu^{''}}}k} \right)}{\overset{\sim}{T}\left( {l^{''}\mu^{''}{{l^{\prime}\mu^{\prime}}}k} \right)}}}}\end{matrix} & (343)\end{matrix}$Now

${\overset{\sim}{V}}_{1}^{(2)}\left( {l\;\mu{{l^{\prime}\mu^{\prime}}}k} \right)$can be expressed in terms of a set of calculable coefficients with asmaller number of unknown and therefore, reduced computational expensefor achieving a given degree of accuracy in the calculated data versesthe actual data. Thus, we can write:

$\begin{matrix}{{{\overset{\sim}{V}}_{1}\left( \overset{\rightarrow}{r} \right)} = {\sum\limits_{\overset{\_}{l}\overset{\_}{\mu}}{Y_{\overset{\_}{l}\overset{\_}{\mu}}{{\overset{\sim}{V}}_{1}\left( {\overset{\_}{l}\overset{\_}{\mu}} \middle| r \right)}}}} & (344)\end{matrix}$

Get Clebsch-Gordan coefficients and a subset of basic integrals, whichgive rise to a formal description of T ⁽¹⁾ (k) as follows:

$\begin{matrix}{{{{\underset{\_}{\underset{\_}{T}}}^{(1)}(k)} = {{{\underset{\_}{\underset{\_}{\overset{\sim}{T}}}}^{(1)}(k)} - {k\;{{\underset{\_}{\underset{\_}{\overset{\sim}{T}}}}^{(1)}(k)}{{\underset{\_}{\underset{\_}{\overset{\sim}{T}}}}^{(2)}(k)}}}}{{{\underset{\_}{\underset{\_}{T}}}^{(2)}(k)} = {{{\underset{\_}{\underset{\_}{\overset{\sim}{T}}}}^{(2)}(k)} - {k\;{{\underset{\_}{\underset{\_}{\overset{\sim}{T}}}}^{(2)}(k)}{{\underset{\_}{\underset{\_}{\overset{\sim}{T}}}}^{(2)}(k)}}}}{{{\underset{\_}{\underset{\_}{T}}}^{(2)}(k)} = {\left\lbrack {\underset{\_}{\underset{\_}{1}} + {k\;{{\underset{\_}{\underset{\_}{\overset{\sim}{T}}}}^{(2)}(k)}}} \right\rbrack^{- 1}{\underset{\_}{\underset{\_}{\overset{\sim}{T}}}}^{(2)}}}} & \; \\\begin{matrix}{{{\underset{\_}{\underset{\_}{T}}}^{(1)}(k)} = {{{\underset{\_}{\underset{\_}{\overset{\sim}{T}}}}^{(1)}(k)}\left\lbrack {\underset{\_}{\underset{\_}{1}} - {{k\left\lbrack {\underset{\_}{\underset{\_}{1}} + {k\;{{\underset{\_}{\underset{\_}{\overset{\sim}{T}}}}^{(2)}(k)}}} \right\rbrack}^{- 1}{{\underset{\_}{\underset{\_}{\overset{\sim}{T}}}}^{(2)}(k)}}} \right\rbrack}} \\{= {{{\underset{\_}{\underset{\_}{\overset{\sim}{T}}}}^{(1)}(k)}\left\lbrack {\underset{\_}{\underset{\_}{1}} + {k\;{{\underset{\_}{\underset{\_}{\overset{\sim}{T}}}}^{(2)}(k)}}} \right\rbrack}^{- 1}}\end{matrix} & \left( {345{a–d}} \right)\end{matrix}$

$\begin{matrix}{{\text{We~~can~~now~~define}\mspace{14mu}{{\underset{\_}{\underset{\_}{V}}}^{(j)}(k)}} = {{\underset{\_}{\underset{\_}{T}}}^{(j)}(k)}} & (346) \\{{\text{and~~we~~get}\mspace{14mu}{{\underset{\_}{\underset{\_}{T}}}^{(i)}(k)}} = {{{\underset{\_}{\underset{\_}{\overset{\sim}{V}}}}^{(1)}(k)}\left\lbrack {\underset{\_}{\underset{\_}{1}} + {k\;{{\underset{\_}{\underset{\_}{\overset{\sim}{V}}}}^{(2)}(k)}}} \right\rbrack}^{- 1}} & (347) \\{\text{This~~give~~us}\mspace{14mu}} & \; \\{{T^{(1)}\left( {l\;\mu{{l^{\prime}\mu^{\prime}}}k} \right)} = {\sum\limits_{l^{''}\mu^{''}}{{{\overset{\sim}{V}}^{(1)}\left( {l\;\mu{{l^{''}\mu^{''}}}k} \right)}\left\{ \left\lbrack {\underset{\_}{\underset{\_}{1}} + {k\;{{\underset{\_}{\underset{\_}{\overset{\sim}{V}}}}^{(2)}(k)}}} \right\rbrack^{- 1} \right\}_{{l\;\mu},{l^{''}{\mu\;}^{''}}}}}} & (348)\end{matrix}$

This expression then is used in the expression for the scatted pressurewave. One determines {tilde over (V)}₁({right arrow over (r)}). Then byusing the full 3D {tilde over (G)}_(0k) operator in the coordinaterepresentation, we obtain the higher order corrections. They will beeasier to determine because of their form than to calculate or determine{tilde over (V)}₁({right arrow over (r)}). These equations have utilityin ultrasound medical imaging and in sonar for submarines and ships toremove reverberations and sonic clutter.

Electromagnetic Inverse Scattering

$\begin{matrix}{{ɛ_{{JM},{J^{\prime}M^{\prime}}}^{{\lambda\lambda}^{\prime}}(r)} = {{{u_{J}({kr})}\delta_{{JJ}^{\prime}}\delta_{{MM}^{\prime}}\delta_{{\lambda\lambda}^{\prime}}} + {\sum\limits_{J^{''}M^{''}\lambda^{\prime\prime\prime}\lambda^{''}}{\int_{0}^{\infty}{{\mathbb{d}r^{\prime}}{\Gamma_{{\lambda\lambda}^{\prime\prime\prime}}^{J +}\left( {r,r^{\prime}} \right)}{\eta_{{JM},{J^{''}M^{''}}}^{\lambda^{\prime\prime\prime}\lambda^{\prime\prime}}\left( r^{\prime} \right)}{ɛ_{{J^{''}M^{''}},{J^{\prime}M^{\prime}}}^{\lambda^{''}\lambda^{\prime}}\left( r^{\prime} \right)}}}}}} & (349) \\{{ɛ\left( {\overset{\rightarrow}{k},{v\;\overset{\rightarrow}{r}}} \right)} = {\frac{4\;\pi}{kr}{\sum\limits_{{JM}\;\lambda\; J^{\prime}M^{\prime}\lambda^{\prime}}{{{\overset{\rightarrow}{Y}}_{JM}^{(\lambda)}\left( \hat{r} \right)}{ɛ_{{JM},{J^{\prime}M^{\prime}}}^{{\lambda\lambda}^{\prime}}(r)}{{{\overset{\rightarrow}{Y}}_{J^{\prime}M^{\prime}}^{(\lambda^{\prime})}\left( \hat{k} \right)} \cdot {{\overset{\rightarrow}{X}}^{\prime}}_{v}}}}}} & (350) \\{{\Gamma_{{\lambda\lambda}^{\prime\prime\prime}}^{J +}\left( {r,r^{\prime}} \right)} = {{{\overset{\sim}{\Gamma}}_{{\lambda\lambda}^{\prime\prime\prime}}^{J}\left( {r,r^{\prime}} \right)} + {{B_{1{\lambda\lambda}^{\prime\prime\prime}}^{J}(r)}{B_{2{\lambda\lambda}^{\prime\prime\prime}}^{J}\left( r^{\prime} \right)}}}} & (351)\end{matrix}$

For the case of λ=e and λ′″=e, we have

$\begin{matrix}{{{{\overset{\sim}{\Gamma}}_{{\lambda\lambda}^{\prime\prime\prime}}^{J}\left( {r,r^{\prime}} \right)} = {k^{2}\left( {{{u_{J}^{\prime}\left( {kr}^{\prime} \right)}{w_{J}^{\prime{( + )}}({kr})}} - {{u_{J}^{\prime}({kr})}{w_{J}^{\prime{( + )}}\left( {kr}^{\prime} \right)}}} \right)}},{{r^{\prime} \leq r} = 0},{r^{\prime} > r}} & (352) \\{{{B_{1{\lambda\lambda}^{\prime\prime\prime}}^{J}(r)}{B_{2{\lambda\lambda}^{\prime\prime\prime}}^{J}\left( r^{\prime} \right)}} = {k^{2}{u_{J}^{\prime}({kr})}{w_{J}^{\prime{( + )}}\left( {kr}^{\prime} \right)}}} & (353)\end{matrix}$

For the case of λ=M and λ′″=M, we have

$\begin{matrix}{{{{\overset{\sim}{\Gamma}}_{MM}^{J}\left( {r,r^{\prime}} \right)} = {k^{2}\left( {{{- {u_{J}^{\prime}\left( {kr}^{\prime} \right)}}{w_{J}^{\prime{( + )}}({kr})}} + {{u_{J}^{\prime}({kr})}{w_{J}^{\prime{( + )}}\left( {kr}^{\prime} \right)}}} \right)}},{{r^{\prime} \leq r} = 0},{r^{\prime} > r}} & (354)\end{matrix}$

$\begin{matrix}{{{B_{1{MM}}^{J}(r)}{B_{2{MM}}^{J}\left( r^{\prime} \right)}} = {{- k^{2}}{u_{J}^{\prime}({kr})}{w_{J}^{\prime{( + )}}\left( {kr}^{\prime} \right)}}} & (355)\end{matrix}$

For the case of λ=0 and λ′″=0, we have

$\begin{matrix}{{{{\overset{\sim}{\Gamma}}_{00}^{J}\left( {r,r^{\prime}} \right)} = {{- \frac{\left( {J + 1} \right)}{{rr}^{\prime}}}\left( {{{u_{J}\left( {kr}^{\prime} \right)}{w_{J}^{( + )}({kr})}} + {{u_{J}({kr})}{w_{J}^{( + )}\left( {kr}^{\prime} \right)}}} \right)}},{{r^{\prime} \leq r} = 0},{r^{\prime} > r}} & (356)\end{matrix}$

$\begin{matrix}{{{B_{100}^{J}(r)}{B_{200}^{J}\left( r^{\prime} \right)}} = {{- \frac{\left( {J + 1} \right)}{{rr}^{\prime}}}{u_{J}({kr})}{w_{J}^{( + )}\left( {kr}^{\prime} \right)}}} & (357)\end{matrix}$and we have

$\begin{matrix}{{{{\overset{\sim}{\Gamma}}_{l\; 0}^{J}\left( {r,r^{\prime}} \right)} = {{{\overset{\sim}{\Gamma}}_{0l}^{J}\left( {r,r^{\prime}} \right)} = {{- \frac{\left\lbrack {J\left( {J + 1} \right)} \right\rbrack^{1/2}}{r^{\prime}}}{\frac{\partial}{\partial r}\left\lbrack {{{u_{J}\left( {kr}^{\prime} \right)}{w_{J}^{( + )}({kr})}} - {{u_{J}({kr})}{w_{J}^{( + )}\left( {kr}^{\prime} \right)}}} \right\rbrack}}}},{{r^{\prime} \leq r} = 0},{r^{\prime} > r}} & (358) \\{{{B_{1l\; 0}^{J}(r)}{B_{20l}^{J}\left( r^{\prime} \right)}} = {{- \frac{\left\lbrack {J\left( {J + 1} \right)} \right\rbrack^{1/2}}{r^{\prime}}}{\frac{\partial}{\partial r}\left\lbrack {{u_{J}({kr})}{w_{J}^{( + )}\left( {kr}^{\prime} \right)}} \right\rbrack}}} & (359) \\{{ɛ_{{JM},{J^{\prime}M^{\prime}}}^{{\lambda\lambda}^{\prime}}(r)} = {{{u_{J}({kr})}\delta_{{JJ}^{\prime}}\delta_{{MM}^{\prime}}{\delta_{{\lambda\lambda}^{\prime}}\left( {\delta_{\lambda\; 0} - 1} \right)}} + {\sum\limits_{\underset{\lambda^{\prime\prime\prime}\lambda^{\prime\prime}}{J^{''}M^{''}}}{{B_{1{\lambda\lambda}^{\prime\prime\prime}}^{J}(r)}{\int_{0}^{\infty}{{\mathbb{d}r^{\prime}}{B_{2{\lambda\lambda}^{\prime\prime\prime}}^{J}\left( r^{\prime} \right)}{\eta_{{JM},{J^{''}M^{''}}}^{\lambda^{\prime\prime\prime}\lambda^{\prime\prime}}\left( r^{\prime} \right)}{ɛ_{{J^{''}M^{''}},{J^{\prime}M^{\prime}}}^{\lambda^{''}\lambda^{\prime}}\left( r^{\prime} \right)}}}}} + {\sum\limits_{\underset{\lambda^{\prime\prime\prime}\lambda^{\prime\prime}}{J^{''}M^{''}}}{\int_{0}^{r}{{\mathbb{d}r^{\prime}}{\Gamma_{{\lambda\lambda}^{\prime\prime\prime}}^{J}\left( {r,r^{\prime}} \right)}{\eta_{{JM},{J^{''}M^{''}}}^{\lambda^{\prime\prime\prime}\lambda^{\prime\prime}}\left( r^{\prime} \right)}{ɛ_{{J^{''}M^{''}},{J^{\prime}M^{\prime}}}^{\lambda^{''}\lambda^{\prime}}\left( r^{\prime} \right)}}}}}} & (360)\end{matrix}$

$\begin{matrix}{{\overset{\leftrightarrow}{ɛ}(r)} = {{{\overset{\leftrightarrow}{U}({kr})} \cdot {\overset{\leftrightarrow}{\Delta}}_{0}} + {{\overset{\leftrightarrow}{U}({kr})} \cdot {\overset{\leftrightarrow}{G}}_{1}} + \mspace{76mu}{\int_{0}^{r}{{\mathbb{d}r^{\prime}}{\overset{\leftrightarrow}{K}\left( {r,r^{\prime}} \right)}{\overset{\leftrightarrow}{ɛ}\left( r^{\prime} \right)}}} + {{\overset{\leftrightarrow}{U}({kr})} \cdot {\overset{\leftrightarrow}{G}}_{2}}}} & (361)\end{matrix}$where

$\begin{matrix}{\left\lbrack {\overset{\leftrightarrow}{ɛ}(r)} \right\rbrack_{\lambda,{JM},{\lambda^{\prime}J^{\prime}M^{\prime}}} = {ɛ_{{JM},{J^{\prime}M^{\prime}}}^{{\lambda\lambda}^{\prime}}(r)}} & (362) \\{\left\lbrack {\overset{\leftrightarrow}{U}(r)} \right\rbrack_{i,j^{\prime}} = {{u_{J}({kr})}\delta_{{JJ}^{\prime}}\delta_{{MM}^{\prime}}\delta_{{\lambda\lambda}^{\prime}}}} & (363) \\{\left\lbrack {\overset{\leftrightarrow}{\Delta}}_{0} \right\rbrack_{{JJ}^{\prime}{MM}^{\prime}{\lambda\lambda}^{\prime}} = {\left( {\delta_{\lambda\; 0} - 1} \right)\delta_{{JJ}^{\prime}}\delta_{{MM}^{\prime}}\delta_{{\lambda\lambda}^{\prime}}}} & (364)\end{matrix}$

Then we can show that

$\begin{matrix}{{\overset{\leftrightarrow}{ɛ}(r)} = {{{{\overset{\leftrightarrow}{ɛ}}_{0}(r)} \cdot \left\lbrack {{\overset{\leftrightarrow}{\Delta}}_{0} + {\overset{\leftrightarrow}{G}}_{1}} \right\rbrack} + {{\overset{\leftrightarrow}{ɛ}\left( r^{\prime} \right)} \cdot {\overset{\leftrightarrow}{G}}_{2}}}} & (365) \\{{{\overset{\leftrightarrow}{ɛ}}_{0}(r)} = {{{\overset{\leftrightarrow}{U}({kr})} + {\int_{o}^{r}{{\mathbb{d}r^{\prime}}{{\overset{\leftrightarrow}{K}\left( {r,r^{\prime}} \right)} \cdot {{\overset{\leftrightarrow}{ɛ}}_{0}\left( r^{\prime} \right)}}}}} = {\sum\limits_{j = 0}^{\infty}{{\overset{\leftrightarrow}{ɛ}}_{0j}(r)}}}} & (366) \\{{{\overset{\leftrightarrow}{ɛ}}_{1}(r)} = {{{\overset{\leftrightarrow}{U}({kr})} + {\int_{o}^{r}{{\mathbb{d}r^{\prime}}{{\overset{\leftrightarrow}{K}\left( {r,r^{\prime}} \right)} \cdot {{\overset{\leftrightarrow}{ɛ}}_{1}\left( r^{\prime} \right)}}}}} = {\sum\limits_{j = 0}^{\infty}{{\overset{\leftrightarrow}{ɛ}}_{1j}(r)}}}} & (367)\end{matrix}$Now asymptotically, for r>r_(max), the scattering amplitude is obtainedfrom

(r) Thus, one determines

η_(JM, J^(′)M^(′))^((1)λλ^(′))(r)from the scattering. Then, one obtains

η_(JM, J^(′)M^(′))^((1)λλ^(′))(r)from the spectra to start the iterative process.

Subtraction Technique Approach to the Inverse Scattering Problem

LetT=V+KT  (368)where

$\begin{matrix}{K = {VG}_{0\; k}^{+}} & (369) \\{G_{0\; k}^{+} = {O_{0\; k} + {\overset{\sim}{G}}_{0k}}} & (370)\end{matrix}$so K=K₁+K₂, whereK₁=VO_(0k)  (371)K₂=V{tilde over (G)}_(0k)  (372)

We now define the following operators as follows:Γ₁=1+K ₁Γ₁  (373)Γ₁=(1+K ₁)⁻¹  (374)Then, we can showT=Γ ₁ V+Γ ₁ K ₂ T=(1−V _(0k))⁻¹ V(1+{tilde over (G)} _(0k) T)  (375)So we define the effective potential {tilde over (V)} as follows:{tilde over (V)}=(1−VO _(0k){)⁻¹ V  (376)

It can be shown that {tilde over (V)} is non-local even if V is local.Therefore, we can show thatT={tilde over (V)}+{tilde over (V)}{tilde over (G)} _(0k) T  (377)We now require that {tilde over (V)} is local (and also that V islocal). Solving for {tilde over (V)}, we obtain{tilde over (V)}=T(1+{tilde over (G)} _(0k) T)⁻¹  (378)Now we perform a power series expansion of the fraction as follows:

$\begin{matrix}{\left( {1 + {{\overset{\sim}{G}}_{0k}T}} \right)^{- 1} = {\sum\limits_{n = 0}^{\infty}\left( {{- {\overset{\sim}{G}}_{0k}}T} \right)^{n}}} & (379)\end{matrix}$to obtain the following equation:

$\begin{matrix}{\overset{\sim}{V} = {\sum\limits_{n = 0}^{\infty}{T\left( {{- {\overset{\sim}{G}}_{0k}}T} \right)}^{n}}} & (380) \\{\mspace{20mu}{\equiv {\sum\limits_{j = 1}^{\infty}{\lambda^{j}{\overset{\sim}{V}}_{j}}}}} & (381)\end{matrix}$and T is first order in λ, so

$\begin{matrix}{{\sum\limits_{j = 1}^{\infty}{\lambda^{j}{\overset{\sim}{V}}_{j}}} = {\sum\limits_{n = 0}^{\infty}{{T\left( {{- {\overset{\sim}{G}}_{0k}}T} \right)}^{n}\lambda^{n + 1}}}} & (382)\end{matrix}$

Then we can write{tilde over (V)}₁=T  (383){tilde over (V)} _(j) =−{tilde over (V)} _(j−1) {tilde over (G)} _(ok){tilde over (V)} ₁  (384)Since {tilde over (V)}₁ must be local in coordinate space, it iscompletely determined by the matrix elements <−{right arrow over(k)}|T|{right arrow over (k)}) (Newton, Scattering Theory of Waves andParticles, Springer-Verlag, New York, 1982, chpt. 20). Additionally,off-shell element of {tilde over (V)}₁ are easily gotten as <{rightarrow over (k)}′|{tilde over (V)}₁|{right arrow over (k)}> so {tildeover (V)}can be constructed. The physical potential is obtained fromequation (376):(1−VO _(0k)){tilde over (V)}=V  (385)so{tilde over (V)}=V+VO _(ok) {tilde over (V)}  (386)

We can evaluate from −k,k matrix elements to solve for {tilde over(V)}(2k) and Fourier invert.V={tilde over (V)}(1+O _(ok) {tilde over (V)})⁻¹  (387)The fact that {tilde over (V)} is local will ensure that V is alsolocal, since O_(0k) is a separable operator. This is an alternative wayto obtain the same inversion as before. The only iteration is for theVolterra kernel equation {tilde over (V)} or T of equation (377). Wealso note thatT={tilde over (V)}+{tilde over (V)}{tilde over (G)} _(k) {tilde over(V)}  (388)The operator {tilde over (G)}_(k) is Volterra so for V local, T is alsoVolterra.

Now evaluating the equations in momentum space, we have(1−VO _(0k)){tilde over (V)}=V  (389)(1+ikπV|k><k|){tilde over (V)}=V  (390){tilde over (V)}(2k)+ikπV(2k){tilde over (V)}(0)=V(2k)  (391)But if we set k=0, then we have{tilde over (V)}(0)=V(0)  (392)that is the average of {tilde over (V)} and V are the same. Therefore,we have{tilde over (V)}(2k)=V(2k)[1−ikπV(0)]  (393)from which {tilde over (V)}(z) can be obtained by inverse Fouriertransformation.

Alternatively,{tilde over (V)}(2k)+ikπV(2k){tilde over (V)}(0)=V(2k)  (394)

$\begin{matrix}{{V\left( {2k} \right)} = \frac{\overset{\sim}{V}\left( {2k} \right)}{1 - {{\mathbb{i}}\; k\;\pi\;{\overset{\sim}{V}(0)}}}} & (395)\end{matrix}$and this expression is correct since V(0) is essentially t(k).

Miscellaneous Volterra Inverse Scattering Results

$\begin{matrix}{\frac{1}{E - K + {\mathbb{i}ɛ}} = {{\overset{\sim}{G}}_{0k} - {\frac{\mathbb{i}\pi}{k}\left. k \right\rangle\left\langle k \right.}}} & (396)\end{matrix}$so we now have

$\begin{matrix}{{\overset{\sim}{G}}_{0k} = {\frac{1}{E - K + {\mathbb{i}ɛ}} + {\frac{\mathbb{i}\pi}{k}\left. k \right\rangle\left\langle k \right.}}} & (397) \\{1 = {\int_{- \infty}^{+ \infty}{{\mathbb{d}r^{\prime}}\left. k^{\prime} \right\rangle\left\langle k^{\prime} \right.}}} & (398) \\{{\delta\left( {E - K} \right)} = {\int_{- \infty}^{+ \infty}{{\mathbb{d}k^{\prime}}{\delta\left( {E - k^{\prime 2}} \right)}\left. k^{\prime} \right\rangle\left\langle k^{\prime} \right.}}} & (399) \\{{E^{\prime} = k^{\prime 2}}{{\mathbb{d}E^{\prime}} = {2k^{\prime}{\mathbb{d}k^{\prime}}}}{{\mathbb{d}k^{\prime}} = {\frac{1}{2\sqrt{E^{\prime}}}{\mathbb{d}E^{\prime}}}}} & \left( {400{a–c}} \right) \\{{\delta\left( {E - K} \right)} = {{\int_{0}^{+ \infty}{{\mathbb{d}k^{\prime}}{\delta\left( {E - k^{\prime 2}} \right)}\left. k^{\prime} \right\rangle\left\langle k^{\prime} \right.}} + \mspace{121mu}{\int_{- \infty}^{0}{{\mathbb{d}k^{\prime}}{\delta\left( {E - k^{\prime 2}} \right)}\left. k^{\prime} \right\rangle\left\langle k^{\prime} \right.}}}} & (401)\end{matrix}$

Now let −u=k′ and du=−k′, then we can derive

$\begin{matrix}{{\delta\left( {E - K} \right)} = {\int_{- \infty}^{+ \infty}{{\mathbb{d}u}\;{\delta\left( {E - u^{2}} \right)}\left. {- u} \right\rangle\left\langle {- u} \right.}}} & (402) \\{{\delta\left( {E - K} \right)} = {\frac{1}{2k}\left\lbrack {{\left. k \right\rangle\left\langle k \right.} + {\left. {- k} \right\rangle\left\langle {- k} \right.}} \right\rbrack}} & (403) \\{{{- {\mathbb{i}}}\;\pi\;{\delta\left( {E - K} \right)}} = {- {\frac{\mathbb{i}\pi}{2\; k}\left\lbrack {{\left. k \right\rangle\left\langle k \right.} + {\left. {- k} \right\rangle\left\langle {- k} \right.}} \right\rbrack}}} & (404)\end{matrix}$Therefore, we can write

$\begin{matrix}{{\overset{\sim}{G}}_{0k} = {\frac{P}{E - K} + {\frac{\mathbb{i}\pi}{2k}\left\lbrack {{\left. k \right\rangle\left\langle k \right.} - {\left. {- k} \right\rangle\left\langle {- k} \right.}} \right\rbrack}}} & (405)\end{matrix}$The principle leading term is part of a Green's function.

General Approach to Volterra-Based Inversion Quantum Scattering Case

We start with

$\begin{matrix}{T = {V + {{VG}_{0k}^{+}T}}} & (406) \\{G_{0k}^{+} = {{- \frac{2{mk}}{\hslash^{2}}}{\sum\limits_{l}{\sum\limits_{m}{{Y_{l\; m}\left( \hat{r} \right)}{Y_{l\; m}\left( {\hat{r}}^{\prime} \right)}^{*}{h_{l}^{+}\left( {kr}_{>} \right)}{j_{l}\left( {kr}_{<} \right)}}}}}} & (407)\end{matrix}$We can now separate

G_(0k)⁺into a Volterra kernel plus a separable Fredholm kernel:

$\begin{matrix}{G_{0k}^{+} = {{\overset{\sim}{G}}_{0k} + O_{k}}} & (408) \\{O_{k} = {{- \frac{2{mk}}{\hslash^{2}}}{\sum\limits_{l}{\sum\limits_{m}{\left. {Y_{l\; m}j_{lk}} \right\rangle\left\langle {Y_{l\; m}h_{lk}^{+}} \right.}}}}} & (409)\end{matrix}$where<{right arrow over (r)}|Y _(lm) j _(lk) >=Y _(lm)({circumflex over (r)})j _(l)(kr)  (410)

$\begin{matrix}\left\langle {{{\overset{\rightharpoonup}{r}}^{\prime}\left. {Y_{l\; m}h_{lk}^{+}} \right\rangle} = {{Y_{l\; m}\left( {\hat{r}}^{\prime} \right)}{h_{lk}^{+}\left( {kr}^{\prime} \right)}}} \right. & (411)\end{matrix}$and with

h_(lk)⁺(kr) = n_(l)(kr) + ij_(l)(kr),

$\begin{matrix}{{G_{0k}^{+} = {{- \frac{2{mk}}{\hslash^{2}}}{\sum\limits_{l}{\sum\limits_{m}\left\lbrack {{\left. {Y_{l\; m}n_{lk}} \right\rangle\left\langle {Y_{l\; m}j_{lk}} \right.} - {\left. {Y_{l\; m}j_{lk}} \right\rangle\left\langle {Y_{l\; m}n_{lk}} \right.}} \right\rbrack}}}},{r^{\prime} \leq r}} & (412) \\{{= 0},{r^{\prime} > r}} & (413)\end{matrix}$

Using equation (407) and equation (410), we can writeT=V+VO _(k) T+V{tilde over (G)} _(0k) T  (414)We can now separate the Volterra and Fredholm pieces of equation (414)by defining {tilde over (T)} byT={tilde over (T)}[1+O _(k) T]  (415){tilde over (T)}=V+V{tilde over (G)} _(0k) {tilde over (T)}  (416)To generate V, we expand it in orders of {tilde over (T)} and requirethe coefficient of each λ^(j) vanish separately. Thus,

$\begin{matrix}{V = {\sum\limits_{j = 1}^{\infty}{\lambda^{J}{\overset{\sim}{V}}_{j}}}} & (417)\end{matrix}$and{tilde over (T)}≈λ{tilde over (T)}  (418)

Then by equation (12)

$\begin{matrix}{{\lambda\;\overset{\sim}{T}} = {{\sum\limits_{j = 1}^{\infty}{\lambda^{j}{\overset{\sim}{V}}_{j}}} + {\sum\limits_{j = 1}^{\infty}{\lambda^{j}{\overset{\sim}{V}}_{j}G_{0k}\lambda\;\overset{\sim}{T}}}}} & (419)\end{matrix}$Therefore{tilde over (V)} ₁={tilde over (T)}  (420)and{tilde over (V)} _(j) =−{tilde over (V)} _(j−1) {tilde over (G)} _(0k){tilde over (V)} ₁ , j>1  (421)

We shall assume that all backscattering elements of T are known, i.e.,all elements of <−{right arrow over (k)}|T|k{right arrow over (k)}> areknown. We now define a local operator V₁({right arrow over (r)}) suchthat

$\begin{matrix}{\left\langle {{- \overset{\rightarrow}{k}}{T}\overset{\rightarrow}{k}} \right\rangle \equiv {\frac{1}{\left( {2\;\pi} \right)^{3}}{\int{{\mathbb{d}{\overset{\rightarrow}{r}}^{\prime}}{\mathbb{e}}^{2{\mathbb{i}}\;{\overset{\rightarrow}{k} \cdot {\overset{\rightarrow}{r}}^{\prime}}}{V_{1}\left( {\overset{\rightarrow}{r}}^{\prime} \right)}}}}} & (422)\end{matrix}$If <−{right arrow over (k)}|T|{right arrow over (k)}> is measured(known), then V₁({right arrow over (r)}) results from the inverseFourier transform as shown below:

$\begin{matrix}{{2{\int{{\mathbb{d}\overset{\rightarrow}{k}}{\mathbb{e}}^{{- 2}{\mathbb{i}}\;{\overset{\rightarrow}{k} \cdot \overset{\rightarrow}{r}}}\left\langle {{- \overset{\rightarrow}{k}}{T}\overset{\rightarrow}{k}} \right\rangle}}} = {\frac{1}{\left( {2\pi} \right)^{3}}{\int{{\mathbb{d}\left( {2\overset{\rightarrow}{k}} \right)}{\int{{\mathbb{d}{\overset{\rightarrow}{r}}^{\prime}}{\mathbb{e}}^{2{\mathbb{i}}\;{\overset{\rightarrow}{k} \cdot {({{\overset{\rightarrow}{r}}^{\prime} - \overset{\rightarrow}{r}})}}}{V_{1}\left( {\overset{\rightarrow}{r}}^{\prime} \right)}}}}}}} & (423)\end{matrix}$which lead to the following:

$\begin{matrix}{{V_{1}\left( \overset{\rightarrow}{r} \right)} = {2{\int{{\mathbb{d}\overset{\rightarrow}{k}}{\mathbb{e}}^{{- 2}{\mathbb{i}}\;{\overset{\rightarrow}{k} \cdot \overset{\rightarrow}{r}}}\left\langle {{- \overset{\rightarrow}{k}}{T}\overset{\rightarrow}{k}} \right\rangle}}}} & (424)\end{matrix}$

We note that once V₁({right arrow over (r)}) is known, then we cancompute its matrix elements in any other representation. To determine{tilde over (V)}₁({right arrow over (r)}), we must solve the Fredholmintegral equation set forth in equation (415). Using equation (410), wehave

$\begin{matrix}{V_{1} = {{\overset{\sim}{V}}_{1}\left\lbrack {1 - {\frac{2{mk}}{\hslash^{2}}{\sum\limits_{l}{\sum\limits_{m}{\left. {Y_{l\; m}j_{lk}} \right\rangle\left\langle {Y_{l\; m}h_{lk}^{+}} \right.V_{1}}}}}} \right\rbrack}} & (425)\end{matrix}$The general matrix element of v, and {tilde over (V)}₁, in the angularmomentum, radial basis are given below:<Y _(lm) j _(lk) |V _(l) |Y _(l′m) ,j _(l′k) >=∫d{right arrow over(r)}∫d{right arrow over (r)}′Y _(lm)* ({circumflex over (r)}) j_(lk)(kr)V ₁({right arrow over (r)},{right arrow over (r)}′)Y _(l′m′)({circumflexover (r)})j_(l′k)(kr′)   (426)But we also haveV ₁({right arrow over (r)},{right arrow over (r)}′)≡δ( {right arrow over(r)}−{right arrow over (r)}′)V ₁({right arrow over (r)})  (427)leading to the following

$\begin{matrix}{\left\langle {Y_{l\; m}j_{lk}{V_{1}}Y_{l^{\prime}m^{\prime}}j_{l^{\prime}k}} \right\rangle = {\int{{\mathbb{d}{\overset{\rightarrow}{r}}^{\prime}}{Y_{l\; m}^{*}\left( \hat{r} \right)}{j_{lk}({kr})}{V_{1}\left( \overset{\rightarrow}{r} \right)}{Y_{l^{\prime}m^{\prime}}\left( \hat{r} \right)}{j_{l^{\prime}k}({kr})}}}} & (428)\end{matrix}$

The same is true for the {tilde over (V)}₁-matrix

$\begin{matrix}{\left\langle {Y_{l\; m}j_{lk}{{\overset{\sim}{V}}_{1}}Y_{l^{\prime}m^{\prime}}j_{l^{\prime}k}} \right\rangle = {\int{{\mathbb{d}{\overset{\rightarrow}{r}}^{\prime}}{Y_{l\; m}^{*}\left( \hat{r} \right)}{j_{lk}({kr})}{{\overset{\sim}{V}}_{1}\left( \overset{\rightarrow}{r} \right)}{Y_{l^{\prime}m^{\prime}}\left( {\hat{r}}^{\prime} \right)}{j_{l^{\prime}k}\left( {kr}^{\prime} \right)}}}} & (429)\end{matrix}$Then equation (425) yields

$\begin{matrix}{\left\langle {Y_{l^{''}m^{''}}j_{l^{''}k}{V_{1}}Y_{l^{\prime}m^{\prime}}j_{l^{\prime}k}} \right\rangle = {\left\langle {Y_{l^{''}m^{''}}j_{l^{''}k}{{\overset{\sim}{V}}_{1}}Y_{l^{\prime}m^{\prime}}j_{l^{\prime}k}} \right\rangle - {\frac{2{mk}}{\hslash^{2}}{\sum\limits_{l}{\sum\limits_{m}{\left\langle {Y_{l^{''}m^{''}}j_{l^{''}k}{{\overset{\sim}{V}}_{1}}Y_{l\; m}j_{lk}} \right\rangle\left\langle {Y_{l\; m}h_{lk}^{+}{V_{1}}Y_{l^{\prime}m^{\prime}}j_{l^{\prime}k}} \right\rangle}}}}}} & (430)\end{matrix}$

This gives a set of linear, inhomogeneous algebraic equations which canbe solved for <Y_(l″m″)j_(l″k)|{tilde over (V)}_(l)|Y_(l′m′) j _(l′k)>.Next, we form

$\begin{matrix}{{\frac{2}{\pi}{\sum\limits_{l^{''}m^{''}}{\sum\limits_{l^{\prime}m^{\prime}}{{Y_{l^{''}m^{''}}\left( {- \hat{k}} \right)}i^{- l^{''}}{Y_{l^{\prime}m^{\prime}}^{*}\left( \hat{k} \right)}\left\langle {Y_{l^{''}m^{''}}j_{l^{''}k}{{\overset{\sim}{V}}_{1}}Y_{l^{\prime}m^{\prime}}j_{l^{\prime}k}} \right\rangle}}}} = \left\langle {{- \overset{\rightarrow}{k}}{{\overset{\sim}{V}}_{1}}\overset{\rightarrow}{k}} \right\rangle} & (431)\end{matrix}$Then we can show that{tilde over (V)} ₁({right arrow over (r)})=2∫d{right arrow over (k)}e^(−2i{right arrow over (k)}·{right arrow over (r)}) <−{right arrow over(k)}|{tilde over (V)} ₁|{right arrow over (k)}>  (432)

Now to generate V_(j) ({right arrow over (r)}), we form the following

$\begin{matrix}{\left\langle {{- \overset{\rightarrow}{k}}{{\overset{\sim}{V}}_{j}}\overset{\rightarrow}{k}} \right\rangle = {- \left\langle {{- \overset{\rightarrow}{k}}{{{\overset{\sim}{V}}_{j - 1}{\overset{\sim}{G}}_{0\; k}{\overset{\sim}{V}}_{1}}}\overset{\rightarrow}{k}} \right\rangle}} & (433) \\{{- \left\langle {{- \overset{\rightarrow}{k}}{{{\overset{\sim}{V}}_{j - 1}{\overset{\sim}{G}}_{0\; k}{\overset{\sim}{V}}_{1}}}\overset{\rightarrow}{k}} \right\rangle} = {- {\int{{\mathbb{d}\overset{\rightarrow}{r}}{\int{{\mathbb{d}{\overset{\rightarrow}{r}}^{\prime}}{\mathbb{e}}^{{\mathbb{i}}\;{\overset{\rightarrow}{k} \cdot \overset{\rightarrow}{r}}}{{\overset{\sim}{V}}_{j - 1}\left( \overset{\rightarrow}{r} \right)}{{\overset{\sim}{G}}_{0\; k}\left( {\overset{\rightarrow}{r},{\overset{\rightarrow}{r}}^{\prime}} \right)}{{\overset{\sim}{V}}_{1}\left( {\overset{\rightarrow}{r}}^{\prime} \right)}{\mathbb{e}}^{{\mathbb{i}}\;{\overset{\rightarrow}{k} \cdot {\overset{\rightarrow}{r}}^{\prime}}}}}}}}} & (434)\end{matrix}$and we also form the following{tilde over (V)} _(j)({right arrow over (r)})=2 ∫d{right arrow over(k)}e ^(−2i{right arrow over (k)}·{right arrow over (r)}) {tilde over(V)} _(j)(2{right arrow over (k)})  (435)This gives a complete, absolutely convergent scheme, combined with asolution of a separable-kernel Fredholm equation of the second kind(inhomogeneous). All the mathematic proofs and derivations are rigorous.

One additional step can be to expand V₁ and {tilde over (V)}₁ inspherical harmonics as follows:

$\begin{matrix}{{{\overset{\sim}{V}}_{1}\left( \overset{\rightarrow}{r} \right)} = {\sum\limits_{\overset{\_}{l}\overset{\_}{m}}{{Y_{\overset{\_}{l}\overset{\_}{m}}\left( \hat{r} \right)}{{\overset{\sim}{V}}_{1}\left( {\overset{\_}{l}\overset{\_}{m}} \middle| r \right)}}}} & (436) \\{{{\overset{\sim}{V}}_{1}\left( \overset{\rightarrow}{r} \right)} = {\sum\limits_{{\overset{-}{l}}^{\prime}{\overset{-}{m}}^{\prime}}{{Y_{{\overset{\_}{l}}^{\prime}{\overset{\_}{m}}^{\prime}}\left( \hat{r} \right)}{{\overset{\sim}{V}}_{1}\left( {{\overset{\_}{l}}^{\prime}{\overset{\_}{m}}^{\prime}} \middle| r \right)}}}} & (437)\end{matrix}$Now we can define

$\begin{matrix}\begin{matrix}{{C\left( {l^{\prime}m^{\prime}\overset{\_}{l}\;\overset{\_}{m}} \middle| {l^{''}m^{''}} \right)} = {\int{{\mathbb{d}\hat{r}}\;{Y_{l^{\prime}m^{\prime}}\left( \hat{r} \right)}{Y_{{\overset{\_}{l}}^{\prime}{\overset{\_}{m}}^{\prime}}\left( \hat{r} \right)}{Y_{l^{''}m^{''}}^{*}\left( \hat{r} \right)}}}} \\{and}\end{matrix} & (438) \\{{{\overset{\sim}{V}}_{1}\left( {\overset{\_}{l}\;\overset{\_}{m}} \middle| {l^{''}l^{\prime}k} \right)} = {\int_{0}^{\infty}\mspace{7mu}{{\mathbb{d}r}\; r^{2}{j_{l^{''}}({kr})}{j_{l^{\prime}}({kr})}{{\overset{\sim}{V}}_{1}\left( {\overset{\_}{l}\;\overset{\_}{m}} \middle| r \right)}}}} & (439) \\{{V_{1}\left( {\overset{\_}{l}\;\overset{\_}{m}} \middle| {l^{''}l^{\prime}k} \right)} = {\int_{0}^{\infty}\mspace{7mu}{{\mathbb{d}r}\; r^{2}{j_{l^{''}}({kr})}{j_{l^{\prime}}({kr})}{V_{1}\left( {\overset{\_}{l}\;\overset{\_}{m}} \middle| r \right)}}}} & (440) \\{{V_{2}\left( {\overset{\_}{l}\;\overset{\_}{m}} \middle| {l^{''}l^{\prime}k} \right)} = {\int_{0}^{\infty}\mspace{7mu}{{\mathbb{d}r}\; r^{2}{h_{l^{''}}({kr})}{j_{l^{\prime}}({kr})}{V_{1}\left( {\overset{\_}{l}\;\overset{\_}{m}} \middle| r \right)}}}} & (441)\end{matrix}$From these results, we get the following

$\begin{matrix}\begin{matrix}{{\sum\limits_{\overset{\_}{l}\;\overset{\_}{m}}^{\;}\;{{C\left( {l^{\prime}m^{\prime}\overset{\_}{l}\;\overset{\_}{m}} \middle| {l^{''}m^{''}} \right)}{V_{1}\left( {\overset{\_}{l}\;\overset{\_}{m}} \middle| {l^{''}l^{\prime}k} \right)}}} =} \\{{\sum\limits_{\overset{\_}{l}\;\overset{\_}{m}}^{\;}\;{{C\left( {l^{\prime}m^{\prime}\overset{\_}{l}\;\overset{\_}{m}} \middle| {l^{''}m^{''}} \right)}{{\overset{\sim}{V}}_{1}\left( {\overset{\_}{l}\;\overset{\_}{m}} \middle| {l^{''}l^{\prime}k} \right)}}} - {\frac{2{mk}}{\hslash^{2}}{\sum\limits_{\overset{\_}{l}\;\overset{\_}{m}}^{\;}{\sum\limits_{{\overset{\_}{l}}^{\prime}\;{\overset{\_}{m}}^{\prime}}^{\;}{\sum\limits_{l\; m}^{\;}{C\left( {l\; m\;\overset{\_}{l}\;\overset{\_}{m}} \middle| {l^{''}m^{''}} \right)}}}}}} \\{{{\overset{\sim}{V}}_{1}\left( {\overset{\_}{l}\;\overset{\_}{m}} \middle| {l^{''}{lk}} \right)}{C\left( {l\; m\;{\overset{\_}{l}}^{\prime}{\overset{\_}{m}}^{\prime}} \middle| {l^{\prime}m^{\prime}} \right)}{V_{2}\left( {{\overset{\_}{l}}^{\prime}{\overset{\_}{m}}^{\prime}} \middle| {l\; l^{\prime}k} \right)}}\end{matrix} & (442)\end{matrix}$Then we want to solve for the elements {tilde over (V)}₁( lm|l″l′k),because there are fewer {tilde over (V)}₁( lm|l″l′k) elements tocalculate than the matrix elements <Y_(l″m″)j_(l″k)|{tilde over(V)}_(l)|Y_(l′m′)j_(l′k)>.

All references cited herein a re incorporated by reference. While thisinvention has been described fully and completely, it should beunderstood that, within the scope of the appended claims, the inventionmay be practiced otherwise than as specifically described. Although theinvention has been disclosed with reference to its preferredembodiments, from reading this description those of skill in the art mayappreciate changes and modification that may be made which do not departfrom the scope and spirit of the invention as described above andclaimed hereafter.

1. A method for analyzing inverse scattering spectral componentscomprising the steps of: irradiating an object with a measuring wave;measuring a reflection spectrum of the object; measuring a transmissionspectrum of the object; calculating a transmission coefficient on acomputer from:${t_{k} = {1 - {\frac{ik}{2}{\int_{- \infty}^{+ \infty}\ {{\mathbb{d}{ze}^{ikz}}{V(z)}{\psi_{k}^{+}(z)}}}}}},$where V(z) is the location interaction between the object and ψ_(k) ⁺(z)is the measuring wave, calculating a reflection coefficient on thecomputer from:$r_{k} = {{- \frac{ik}{2}}{\int_{- \infty}^{- \infty}{e^{- {ikz}}{V(z)}{\psi_{k}^{+}(z)}}}}$using a set of definitions${t_{k}{{\overset{\sim}{\psi}}_{k}(z)}} = {\psi_{k}^{+}(z)}$$\frac{r_{k}}{t_{k}} = {\overset{\sim}{r}}_{k}$${{\overset{\sim}{V}}_{1}(z)} = {\int_{- \infty}^{+ \infty}\ {{\mathbb{d}\left( {2\; k} \right)}\frac{2\; i}{k}{\overset{\sim}{r}}_{k}e^{{- 2}{ikz}}}}$to convert a Lippmann-Schwinger inverse scattering equation${\psi_{k}^{+}(z)} = {e^{ikz} - {\frac{ik}{2}{\int_{- \infty}^{+ \infty}\ {{\mathbb{d}z^{\prime}}e^{{ik}{{z - z^{\prime}}}}{V\left( z^{\prime} \right)}{\psi_{k}^{+}\left( z^{\prime} \right)}}}}}$on the computer in a Volterra-type form${{{\overset{\sim}{V}}_{1}(z)} = {\int_{- \infty}^{+ \infty}\ {{\mathbb{d}\left( {2k} \right)}{\mathbb{e}}^{{- 2}\;{\mathbb{i}}\;{kz}}\frac{2i}{k}{r_{k}\left\lbrack {1 + {\frac{i\; k\;\Delta}{2}{\sum\limits_{j}^{\;}\;{{\mathbb{e}}^{{- {\mathbb{i}}}\; k\; z_{j}}{V\left( z_{j} \right)}{{\overset{\sim}{\psi}}_{k}(z)}}}}} \right\rbrack}}}};$and iterating the Volterra-form of the Lippmann-Schwinger equation onthe computer to produce an approximate solution {tilde over (V)}₁(z),where {tilde over (V)}₁(z) is absolutely and uniformly convergent. 2.The method of claim 1, wherein the approximate solution {tilde over(V)}₁(z) includes four terms.
 3. The method of claim 1, wherein theapproximate solution {tilde over (V)}₁(z) includes three terms.
 4. Themethod of claim 1, wherein the approximate solution {tilde over (V)}₁(z)includes two terms.
 5. A method for analyzing inverse scatteringcomponents of a spectrum of an object of interest comprising the stepsof: obtaining a reflectance and/or transmission spectra of an object ofinterest using an incident waveform from the group consisting of anelectromagnetic waveform, sonic waveform and mixtures or combinationsthereof; claculating a transmission coefficient on a computer from:${t_{k} = {1 - {\frac{ik}{2}{\int_{- \infty}^{+ \infty}\ {{\mathbb{d}{ze}^{ikz}}{V(z)}{\psi_{k}^{+}(z)}}}}}},$where V(z) is the location interaction between the object and ψ_(k) ⁺(z)is the measuring wave, calculating a reflection coefficient on thecomputer from:$r_{k} = {{- \frac{ik}{2}}{\int_{- \infty}^{+ \infty}{e^{- {ikz}}{V(z)}{\psi_{k}^{+}(z)}}}}$using a set of definitions${t_{k}{{\overset{\sim}{\psi}}_{k}(z)}} = {\psi_{k}^{+}(z)}$$\frac{r_{k}}{t_{k}} = {\overset{\sim}{r}}_{k}$${{\overset{\sim}{V}}_{1}(z)} = {\int_{- \infty}^{+ \infty}\ {{\mathbb{d}\left( {2\; k} \right)}\frac{2\; i}{k}{\overset{\sim}{r}}_{k}e^{{- 2}{ikz}}}}$to convert a Lippmann-Schwinger inverse scattering equation${\psi_{k}^{+}(z)} = {e^{ikz} - {\frac{ik}{2}{\int_{- \infty}^{+ \infty}\ {{\mathbb{d}z^{\prime}}e^{{ik}{{z - z^{\prime}}}}{V\left( z^{\prime} \right)}{\psi_{k}^{+}\left( z^{\prime} \right)}}}}}$on the computer into a Volterra-type form${{{\overset{\sim}{V}}_{1}(z)} = {\int_{- \infty}^{+ \infty}\ {{\mathbb{d}\left( {2k} \right)}e^{{- 2}{ikz}}\frac{2\; i}{k}{r_{k}\left\lbrack {1 + {\frac{ik}{2}{\int_{- \infty}^{+ \infty}{e^{- {ikz}}{V\left( z_{j} \right)}{{\overset{\sim}{\psi}}_{k}(z)}}}}} \right\rbrack}}}};$and iterating the Volterra-type form of the Lippmann-Schwinger equationon the computer to produce {tilde over (V)}₁(z), where {tilde over(V)}₁(z) is absolutely and uniformly convergent.
 6. The method of claim5, wherein the approximate solution {tilde over (V)}₁(z) includes fourterms.
 7. The method of claim 5, wherein the approximate solution {tildeover (V)}₁(z) includes three terms.
 8. The method of claim 5, whereinthe approximate solution {tilde over (V)}₁(z) includes two terms.
 9. Ananalytical instrument including an excitation source for producing anincident waveform, a detector for receiving either a transmissionspectrum or a reflectance spectrum or both a transmission spectrum and areflectance spectrum of an object or volume of interest, and aprocessing unit for analyzing the spectra, where the processing unitincludes software encoding the inverse scattering method of claims 1, 2,3, 4, 5, 6, 7, or
 8. 10. A sonic analytical instrument including a sonicexcitation source for producing an incident sonic waveform, a detectorfor receiving either a sonic transmission spectrum or a sonicreflectance spectrum or both a sonic transmission spectrum and a sonicreflectance spectrum of an object or volume of interest, and aprocessing unit for analyzing the sonic spectra, where the processingunit includes software encoding the inverse scattering method of claims1, 2, 3, 4, 5, 6, 7, or
 8. 11. An electromagnetic analytical instrumentincluding an electromagnetic excitation source for producing an incidentelectromagnetic waveform, a detector for receiving either anelectromagnetic transmission spectrum or an electromagnetic reflectancespectrum or both an electromagnetic transmission spectrum and anelectromagnetic reflectance spectrum of an object or volume of interest,and a processing unit for analyzing the electromagnetic spectra, wherethe processing unit includes software encoding the inverse scatteringmethod of claims 1, 2, 3, 4, 5, 6, 7, or
 8. 12. An analytical instrumentincluding a sonic excitation source and an electromagnetic excitationsource for producing an incident sonic waveform and an incidentelectromagnetic waveform, a detector for receiving either a sonictransmission spectrum or a sonic reflectance spectrum or both a sonictransmission spectrum and a sonic reflectance spectrum of an object orvolume of interest, a detector for receiving either an electromagnetictransmission spectrum or an electromagnetic reflectance spectrum or bothan electromagnetic transmission spectrum and an electromagneticreflectance spectrum of an object or volume of interest, and aprocessing unit for analyzing the sonic and electromagnetic spectra,where the processing unit includes software encoding the inversescattering method of claims 1, 2, 3, 4, 5, 6, 7, or 8.